Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic Mechanics
LLINEAR ALMOST POISSON STRUCTURES AND HAMILTON-JACOBI EQUATION.APPLICATIONS TO NONHOLONOMIC MECHANICS
MANUEL DE LE ´ON, JUAN C. MARRERO, AND DAVID MART´IN DE DIEGO
Abstract.
In this paper, we study the underlying geometry in the classical Hamilton-Jacobi equation. The proposedformalism is also valid for nonholonomic systems. We first introduce the essential geometric ingredients: a vectorbundle, a linear almost Poisson structure and a Hamiltonian function, both on the dual bundle (a Hamiltonian system).From them, it is possible to formulate the Hamilton-Jacobi equation, obtaining as a particular case, the classical theory.The main application in this paper is to nonholonomic mechanical systems. For it, we first construct the linear almostPoisson structure on the dual space of the vector bundle of admissible directions, and then, apply the Hamilton-Jacobitheorem. Another important fact in our paper is the use of the orbit theorem to symplify the Hamilton-Jacobi equation,the introduction of the notion of morphisms preserving the Hamiltonian system; indeed, this concept will be very usefulto treat with reduction procedures for systems with symmetries. Several detailed examples are given to illustrate theutility of these new developments.
Contents
1. Introduction 22. Linear almost Poisson structures, skew-symmetric algebroids and almost differentials 43. Skew-symmetric algebroids and the orbit theorem 94. Linear almost Poisson structures and Hamilton-Jacobi equation 124.1. Linear almost Poisson structures and Hamiltonian systems 124.2. Hamiltonian systems and Hamilton-Jacobi equation 134.3. Linear almost Poisson morphisms and Hamilton-Jacobi equation 165. Applications to nonholonomic Mechanics 195.1. Unconstrained mechanical systems on a Lie algebroid 195.2. Mechanical systems subjected to linear nonholonomic constraints on a Lie algebroid 256. Conclusions and Future Work 39Appendix 39References 41
Mathematics Subject Classification.
Key words and phrases.
Hamilton-Jacobi equation, linear almost Poisson structure, almost differential, skew-symmetric algebroid, orbittheorem, Hamiltonian morphism, nonholonomic mechanical system.This work has been partially supported by MEC (Spain) Grants MTM 2006-03322, MTM 2007-62478, project ”Ingenio Mathematica”(i-MATH) No. CSD 2006-00032 (Consolider-Ingenio 2010) and S-0505/ESP/0158 of the Comunidad de Madrid. We gratefully acknowledgehelpful comments and suggestions of Anthony Bloch, Eduardo Mart´ınez and Tomoki Oshawa. The authors also thank the referees, whosuggested important improvements upon the first versions of our paper. a r X i v : . [ m a t h - ph ] N ov M. DE LE ´ON, J. C. MARRERO, AND D. MART´IN DE DIEGO Introduction
The standard Hamilton-Jacobi equation is the first-order, non-linear partial differential equation, ∂S∂t + H ( q A , ∂S∂q A ) = 0 , (1.1)for a function S ( t, q A ) (called the principal function) and where H is the Hamiltonian function of thesystem. Taking S ( t, q A ) = W ( q A ) − tE , where E is a constant, we rewrite the previous equations as H ( q A , ∂W∂q A ) = E, (1.2)where W is called the characteristic function. Equations (1.1) and (1.2) are indistinctly referred asthe Hamilton-Jacobi equation (see [1, 11]; see also [6] for a recent geometrical approach).The motivation of the present paper is to extend this theory for the case of nonholonomic me-chanical systems, that is, those mechanical systems subject to linear constraints on the velocities. InRemark 5.11 of our paper, we carefully summarize previous approaches to this subject. These triedto adapt the standard Hamilton-Jacobi equations for systems without constraints to the nonholo-nomic setting. But for nonholonomic mechanics is necessary to take into account that the dynamicsis obtained from an almost Poisson bracket, that is, a bracket not satisfying the Jacobi identity. Inthis direction, in a recent paper [20], the authors have developed a new approach which permits toextend the Hamilton-Jacobi equation to nonholonomic mechanical systems. However, the expressionof the corresponding Hamilton-Jacobi equation is far from the standard Hamilton-Jacobi equationfor unconstrained systems. This fact has motivated the present discussion since it was necessaryto understand the underlying geometric structure in the proposed Hamilton-Jacobi equation fornonholonomic systems.To go further in this direction, we need a new framework, which captures the non-Hamiltonianessence of a nonholonomic problem. Thus, we have considered a more general minimal “Hamiltonian”scenario. The starting point is a vector bundle τ D : D −→ Q such that its dual vector bundle τ D ∗ : D ∗ −→ Q is equipped with a linear almost Poisson bracket {· , ·} D ∗ , that is, a linear bracketsatisfying all the usual properties of a Poisson bracket except the Jacobi identity. The existence ofsuch bracket is equivalent to the existence of an skew-symmetric algebroid structure ([[ · , · ]] D , ρ D ) on τ D : D −→ Q (i.e. a Lie algebroid structure eliminating the integrability property), or even, theexistence of an almost differential d D on τ D : D −→ Q , that is, an operator d D which acts on the“forms” on D and it satisfies all the properties of an standard differential except that ( d D ) is not, ingeneral, zero. We remark that skew-symmetric algebroid structures are almost Lie structures in theterminology of [34] (see also [35]) and that the one-to-one correspondence between skew-symmetricalgebroids and almost differentials was obtained in [34]. We also note that an skew-symmetricalgebroid also is called a pre-Lie algebroid in the terminology introduced in some papers (see, forinstance, [15, 16, 23]) and the relation between linear almost Poisson brackets and skew-symmetricalgebroid structures was discussed in these papers (see also [12, 13] for some applications to ClassicalMechanics).In this framework, a Hamiltonian system is given by a Hamiltonian function h : D ∗ −→ R . Thedynamics is provided by the corresponding Hamiltonian vector field H Λ D ∗ h ( H Λ D ∗ h ( f ) = { f, h } D ∗ ,for all real function f on D ∗ ). Here, Λ D ∗ is the almost Poisson tensor field defined from {· , ·} D ∗ .The reader can immediately recognize that we are extending the standard model, where D = T Q , AMILTON-JACOBI EQUATION AND NONHOLONOMIC MECHANICS 3 D ∗ = T ∗ Q and [[ · , · ]] D is the usual Lie bracket of vector fields which is related with the canonicalPoisson bracket on T ∗ Q , so that d D is just, in this case, the usual exterior differential. Anotherimportant fact is the introduction of the notion of morphisms preserving the Hamiltonian system;indeed, this concept will be very useful to treat with reduction procedures for nonholonomic systemswith symmetries. We remark that this type of procedures were intensively discussed in the seminalpaper [3] by Bloch et al. In the above framework we can prove the main result of our paper: Theorem 4.1. In this theorem,we obtain the Hamilton-Jacobi equation whose expression seems a natural extension of the classicalHamilton-Jacobi equation for unconstrained systems, as appears, for instance, in [1]. Moreover, ourconstruction is preserved under the natural morphims of the theory. This fact is proved in Theorem4.12.Furthermore, using the orbit theorem (see [2]), we will show that the classical form of the Hamilton-Jacobi equation: H ◦ α = constant, with α : Q → D ∗ satisfying d D α = 0, remains valid for aspecial class of nonholonomic mechanical systems: those satisfying the condition of being completelynonholonomic. See Section 3 for more details and also the paper by Ohsawa and Bloch [31] for theparticular case when D is a distribution on Q .The above theorems are applied to the theory of mechanical systems subjected to linear nonholo-nomic constraints on a Lie algebroid A . The ingredients of this theory are a Lie algebroid τ A : A → Q over a manifold Q , a Lagrangian function L : A → R of mechanical type, and a vector subbundle τ D : D → Q of A . The total space D of this vector subbundle is the constraint submanifold (see[7]). Then, using the corresponding linear Poisson structure on A ∗ , one may introduce a linear al-most Poisson bracket on D ∗ , the so-called nonholonomic bracket. A linear almost Poisson bracketon D which is isomorphic to the nonholonomic bracket was considered in [7]; however, it shouldbe remarked that our formalism simplifies very much the procedure to obtain it. Using all theseingredients one can apply the general procedure (Theorems 4.1 and 4.12) to obtain new and inter-esting results. We also remark that the main part of the relevant information for developing theHamilton-Jacobi equation for the nonholonomic system ( L, D ) is contained in the vector subbundle D or, equivalently, in its dual D ∗ (see Theorems 4.1 and 4.12). Then, the computational cost is lowerthan in previous approximations to the theory.In the particular case when A is the standard Lie algebroid τ T Q : T Q → Q then the constraintsubbundle is a distribution D on Q . The linear almost Poisson bracket on D ∗ is provided by theclassical nonholonomic bracket (which is usually induced from the canonical Poisson bracket on T ∗ Q ),clarifying previous constructions [5, 19, 39]. In addition, as a consequence, we recover some of theresults obtained in [20] about the Hamilton-Jacobi equation for nonholonomic mechanical systems(see Corollary 5.9). Moreover, we apply our results to an explicit example: the two-wheeled carriage.On the other hand, if our Lagrangian system on an arbitrary Lie algebroid A is unconstrained(that is, the constraint subbundle D = A ) then, using our general theory, we recover some resultson the Hamilton-Jacobi equation for Lie algebroids (see Corollary 5.1) which were proved in [25].Furthermore, if A is the standard Lie algebroid τ T Q : T Q → Q then we directly deduce somewell-known facts about the classical Hamilton-Jacobi equation (see Corollary 5.2).Another interesting application is discussed; the particular case when the Lie algebroid is theAtiyah algebroid τ ¯ A : ¯ A = T Q/G → ¯ Q = Q/G associated with a principal G -bundle F : Q → ¯ Q = Q/G . In such a case, we have a Lagrangian function ¯ L : ¯ A → R of mechanical type and aconstraint subbundle τ ¯ D : ¯ D → ¯ Q of τ ¯ A : ¯ A = T Q/G → ¯ Q . This nonholonomic system is precisely M. DE LE ´ON, J. C. MARRERO, AND D. MART´IN DE DIEGO the reduction, in the sense of Theorem 4.12, of a constrained system (
L, D ) on the standard Liealgebroid τ A : A = T Q → Q . In fact, using Theorem 4.12, we deduce that the solutions of theHamilton-Jacobi equations for both systems are related in a natural way by projection. We alsocharacterize the nonholonomic bracket on ¯ D ∗ . All these results are applied to a very interestingexample: the snakeboard. In this example, an explicit expression of the reduced nonholonomicbracket is found; moreover, the Hamilton-Jacobi equations are proposed and it is shown the utilityof our framework to integrate the equations of motion.We expect that the results of this paper will be useful for analytical integration of many difficultsystems (see, as an example, the detailed study of the snakeboard in this paper and the examplesin [31]) and the key for the construction of geometric integrators based on the Hamilton-Jacobiequation (see, for instance, Chapter VI in [18] and references therein for the particular case ofstandard nonholonomic mechanical systems).The structure of the paper is as follows. In Section 2, the relation between linear almost Poissonstructures on a vector bundle, skew-symmetric algebroids and almost differentials is obtained. InSection 3, we introduce the notion of a completely nonholonomic skew-symmetric algebroid andwe prove that on an algebroid D of this kind with connected base Q the space H ( d D ) = { f ∈ C ∞ ( Q ) /d D f = 0 } is isomorphic to R . We also prove that on an arbitrary skew-symmetric algebroid D the condition d D f = 0 implies that f is constant on the leaves of a certain generalized foliation(see Theorem 3.4). For this purpose, we will use the orbit theorem. In Section 4, we considerHamiltonian systems associated with a linear almost Poisson structure on the dual bundle D ∗ to avector bundle and a Hamiltonian function on D ∗ . Then, the Hamilton-Jacobi equation is proposedin this setting. Moreover, using the results of Section 3, we obtain an interesting expression of thisequation. In Section 5, we apply the previous results to nonholonomic mechanical systems and, inparticular, to some explicit examples. Moreover, we review in this section some previous approachesto the topic. We conclude our paper with the future lines of work and an appendix with the proofof some technical results.2. Linear almost Poisson structures, skew-symmetric algebroids and almostdifferentials
Most of the results contained in this section are well-known in the literature (see [14, 15, 16, 34, 35]).However, to make the paper more self-contained, we will include their proofs.Let τ D : D → Q be a vector bundle of rank n over a manifold Q of dimension m . Denote by D ∗ the dual vector bundle to D and by τ D ∗ : D ∗ → Q the corresponding vector bundle projection. Definition 2.1. A linear almost Poisson structure on D ∗ is a bracket of functions {· , ·} D ∗ : C ∞ ( D ∗ ) × C ∞ ( D ∗ ) → C ∞ ( D ∗ ) such that: (i) {· , ·} D ∗ is skew-symmetric, that is, { ϕ, ψ } D ∗ = −{ ψ, ϕ } D ∗ , for ϕ, ψ ∈ C ∞ ( D ∗ ) . (ii) {· , ·} D ∗ satisfies the Leibniz rule, that is, { ϕϕ (cid:48) , ψ } D ∗ = ϕ { ϕ (cid:48) , ψ } D ∗ + ϕ (cid:48) { ϕ, ψ } D ∗ , for ϕ, ϕ (cid:48) , ψ ∈ C ∞ ( D ∗ ) . AMILTON-JACOBI EQUATION AND NONHOLONOMIC MECHANICS 5 (iii) {· , ·} D ∗ is linear, that is, if ϕ and ψ are linear functions on D ∗ then { ϕ, ψ } D ∗ is also a linearfunction.If, in addition, the bracket satisfies the Jacobi identity then we have that {· , ·} D ∗ is a linearPoisson structure on D ∗ . Properties (i) and (ii) in Definition 2.1 imply that there exists a 2-vector Λ D ∗ on D ∗ such thatΛ D ∗ ( dϕ, dψ ) = { ϕ, ψ } D ∗ , for ϕ, ψ ∈ C ∞ ( D ∗ ) . Λ D ∗ is called the linear almost Poisson -vector associated with the linear almost Poissonstructure {· , ·} D ∗ .Note that there exists a one-to-one correspondence between the space Γ( τ D ) of sections of thevector bundle τ D : D → Q and the space of linear functions on D ∗ . In fact, if X ∈ Γ( τ D ) then thecorresponding linear function ˆ X on D ∗ is given byˆ X ( α ) = α ( X ( τ D ∗ ( α ))) , for α ∈ D ∗ . Proposition 2.2.
Let {· , ·} D ∗ be a linear almost Poisson structure on D ∗ . (i) If X is a section of τ D : D → Q and f is a real C ∞ -function on Q then the bracket { ˆ X, f ◦ τ D ∗ } D ∗ is a basic function with respect to the projection τ D ∗ . (ii) If f and g are real C ∞ -functions on Q then { f ◦ τ D ∗ , g ◦ τ D ∗ } D ∗ = 0 . Proof.
Let Y be an arbitrary section of τ D : D → Q .Using Definition 2.1, we have that { ˆ X, ( f ◦ τ D ∗ ) ˆ Y } D ∗ = ( f ◦ τ D ∗ ) { ˆ X, ˆ Y } D ∗ + ( { ˆ X, f ◦ τ D ∗ } D ∗ ) ˆ Y is a linear function on D ∗ . Thus, since ( f ◦ τ D ∗ ) { ˆ X, ˆ Y } D ∗ also is a linear function, it follows that { ˆ X, f ◦ τ D ∗ } D ∗ is a basic function with respect to τ D ∗ . This proves (i).On the other hand, using (i) and Definition 2.1, we deduce that { ( f ◦ τ D ∗ ) ˆ Y , g ◦ τ D ∗ } D ∗ = ( f ◦ τ D ∗ ) { ˆ Y , g ◦ τ D ∗ } D ∗ + ( { f ◦ τ D ∗ , g ◦ τ D ∗ } D ∗ ) ˆ Y is a basic function with respect to τ D ∗ . Therefore, as ( f ◦ τ D ∗ ) { ˆ Y , g ◦ τ D ∗ } D ∗ also is a basic functionwith respect to τ D ∗ , we conclude that { f ◦ τ D ∗ , g ◦ τ D ∗ } D ∗ = 0. This proves (ii). (cid:3) If ( q i ) are local coordinates on an open subset U of Q and { X α } is a basis of sections of the vectorbundle τ − D ( U ) → U then we have the corresponding local coordinates ( q i , p α ) on D ∗ . Moreover, fromProposition 2.2, it follows that { p α , p β } D ∗ = − C γαβ p γ , { q j , p α } D ∗ = ρ jα , { q i , q j } D ∗ = 0 , with C γαβ and ρ jα real C ∞ -functions on U .Consequently, the linear almost Poisson 2-vector Λ D ∗ has the following local expressionΛ D ∗ = ρ jα ∂∂q j ∧ ∂∂p α − C γαβ p γ ∂∂p α ∧ ∂∂p β . (2.1) M. DE LE ´ON, J. C. MARRERO, AND D. MART´IN DE DIEGO
Definition 2.3. An skew-symmetric algebroid structure on the vector bundle τ D : D → Q isa R -linear bracket [[ · , · ]] D : Γ( τ D ) × Γ( τ D ) → Γ( τ D ) on the space Γ( τ D ) and a vector bundle morphism ρ D : D → T Q , the anchor map , such that: (i) [[ · , · ]] D is skew-symmetric, that is, [[ X, Y ]] D = − [[ Y, X ]] D , for X, Y ∈ Γ( τ D ) . (ii) If we also denote by ρ D : Γ( τ D ) → X ( Q ) the morphism of C ∞ ( Q ) -modules induced by theanchor map then [[ X, f Y ]] D = f [[ X, Y ]] D + ρ D ( X )( f ) Y, for X, Y ∈ Γ( D ) and f ∈ C ∞ ( Q ) . If the bracket [[ · , · ]] D satisfies the Jacobi identity, we have that the pair ([[ · , · ]] D , ρ D ) is a Lie alge-broid structure on the vector bundle τ D : D → Q . Remark 2.4.
If ( D, [[ · , · ]] D , ρ D ) is a Lie algebroid over Q , we may consider the generalized distribution˜ D whose characteristic space at a point q ∈ Q is given by ˜ D ( q ) = ρ D ( D q ), where D q is the fibreof D over q . The distribution ˜ D is finitely generated and involutive. Thus, ˜ D defines a generalizedfoliation on Q in the sense of Sussmann [38]. ˜ D is the Lie algebroid foliation on Q associatedwith D . (cid:5) Now, we will denote by
LAP ( D ∗ ) (respectively, LP ( D ∗ )) the set of linear almost Poisson structures(respectively, linear Poisson structures) on D ∗ . Denote also by SSA ( D ) (respectively, LA ( D )) theset of skew-symmetric algebroid (respectively, Lie algebroid) structures on the vector bundle τ D : D → Q . Then, we will see in the next theorem that there exists a one-to-one correspondencebetween LAP ( D ∗ ) (respectively, LP ( D ∗ )) and the set of skew-symmetric algebroid (respectively, Liealgebroid) structures on τ D : D → Q . Theorem 2.5.
There exists a one-to-one correspondence Ψ between the sets LAP ( D ∗ ) and SSA ( D ) .Under the bijection Ψ , the subset LP ( D ∗ ) of LAP ( D ∗ ) corresponds with the subset LA ( D ) of SSA ( D ) . Moreover, if {· , ·} D ∗ is a linear almost Poisson structure on D ∗ then the correspondingskew-symmetric algebroid structure ([[ · , · ]] D , ρ D ) on D is characterized by the following conditions (cid:92) [[ X, Y ]] D = −{ ˆ X, ˆ Y } D ∗ , ρ D ( X )( f ) ◦ τ D ∗ = { f ◦ τ D ∗ , ˆ X } D ∗ (2.2) for X, Y ∈ Γ( τ D ) and f ∈ C ∞ ( Q ) .Proof. Let {· , ·} D ∗ be a linear almost Poisson structure on D ∗ . Then, it is easy to prove that [[ · , · ]] D (defined as in (2.2)) is a R -bilinear skew-symmetric bracket. Moreover, since {· , ·} D ∗ satisfies theLeibniz rule, it follows that ρ D ( X ) is a vector field on Q , for X ∈ Γ( τ D ). In addition, using againthat {· , ·} D ∗ satisfies the Leibniz rule and Proposition 2.2, we deduce that ρ D ( gX ) = gρ D ( X ) , for g ∈ C ∞ ( Q ) and X ∈ Γ( τ D ) . Thus, ρ D : Γ( τ D ) → X ( Q ) is a morphism of C ∞ ( Q )-modules.On the other hand, from (2.2), we obtain that (cid:92) [[ X, f Y ]] D = −{ ˆ X, ( f ◦ τ D ∗ ) ˆ Y } D ∗ = ( ρ D ( X )( f ) ◦ τ D ∗ ) ˆ Y − ( f ◦ τ D ∗ ) { ˆ X, ˆ Y } D ∗ . Therefore, [[
X, f Y ]] D = f [[ X, Y ]] D + ρ D ( X )( f ) Y AMILTON-JACOBI EQUATION AND NONHOLONOMIC MECHANICS 7 and ([[ · , · ]] D , ρ D ) is an skew-symmetric algebroid structure on τ D : D → Q . It is clear that if {· , ·} D ∗ is a Poisson bracket then [[ · , · ]] D satisfies the Jacobi identity.Conversely, if ([[ · , · ]] D , ρ D ) is an skew-symmetric algebroid structure on τ D : D → Q and x ∈ Q then one may prove that there exists an open subset U of Q and a unique linear almost Poissonstructure on the vector bundle τ τ − D ( U ) ∗ : τ − D ( U ) ∗ → U such that { ˆ X, ˆ Y } τ − D ( U ) ∗ = − (cid:92) [[ X, Y ]] τ − D ( U ) , { f ◦ τ τ − D ( U ) ∗ , ˆ X } τ − D ( U ) ∗ = ρ τ − D ( U ) ( X )( f ) ◦ τ τ − D ( U ) ∗ , and { f ◦ τ τ − D ( U ) ∗ , g ◦ τ τ − D ( U ) ∗ } τ − D ( U ) ∗ = 0 , for X, Y sections of the vector bundle τ − D ( U ) → U and f, g ∈ C ∞ ( U ). Here, ([[ · , · ]] τ − D ( U ) , ρ τ − D ( U ) ) isthe skew-symmetric algebroid structure on τ − D ( U ) induced, in a natural way, by the skew-symmetricalgebroid structure ([[ · , · ]] D , ρ D ) on D . In addition, we have that if [[ · , · ]] D satisfies the Jacobi identitythen {· , ·} τ − D ( U ) ∗ is a linear Poisson bracket on τ − D ( U ) ∗ . Thus, we deduce that there exists a uniquelinear almost Poisson structure {· , ·} D ∗ on D ∗ such that (2.2) holds. (cid:3) Let {· , ·} D ∗ be a linear almost Poisson structure on D ∗ and ([[ · , · ]] D , ρ D ) be the corresponding skew-symmetric algebroid structure on τ D : D → Q . If ( q i ) are local coordinates on an open subset U of Q and { X α } is a basis of sections of the vector bundle τ − D ( U ) → U such that Λ D ∗ is given by (2.1)(on τ − D ( U )) then [[ X α , X β ]] D = C γαβ X γ , ρ D ( X α ) = ρ jα ∂∂q j .C γαβ and ρ jα are called the local structure functions of the skew-symmetric algebroid structure([[ · , · ]] D , ρ D ) with respect to the local coordinates ( q i ) and the basis { X α } .Next, we will see that there exists a one-to-one correspondence between SSA ( D ) and the set ofalmost differentials on the vector bundle τ D : D → M . Definition 2.6.
An almost differential on the vector bundle τ D : D → Q is a R -linear map d D : Γ(Λ k τ D ∗ ) → Γ(Λ k +1 τ D ∗ ) , k ∈ { , . . . , n − } such that d D ( α ∧ β ) = d D α ∧ β + ( − k α ∧ d D β, for α ∈ Γ(Λ k τ D ∗ ) and β ∈ Γ(Λ r τ D ∗ ) . (2.3) If ( d D ) = 0 then d D is said to be a differential on the vector bundle τ D : D → Q . Denote by AD ( D ) (respectively, D ( D )) the set of almost differentials (respectively, differentials)on the vector bundle τ D : D → Q . Theorem 2.7.
There exists a one-to-one correspondence Φ between the sets SSA ( D ) and AD ( D ) .Under the bijection Φ the subset LA ( D ) of SSA ( D ) corresponds with the subset D ( D ) of AD ( D ) .Moreover, we have: (i) If d D is an almost differential on the vector bundle τ D : D → Q then the corresponding skew-symmetric algebroid structure ([[ · , · ]] D , ρ D ) on D is characterized by the following conditions: α ([[ X, Y ]] D ) = d D ( α ( Y ))( X ) − d D ( α ( X ))( Y ) − ( d D α )( X, Y ) , ρ D ( X )( f ) = ( d D f )( X ) , (2.4) for X, Y ∈ Γ( τ D ) , α ∈ Γ( τ D ∗ ) and f ∈ C ∞ ( Q ) . M. DE LE ´ON, J. C. MARRERO, AND D. MART´IN DE DIEGO (ii) If ([[ · , · ]] D , ρ D ) is an skew-symmetric algebroid structure on the vector bundle τ D : D → Q then the corresponding almost differential d D is defined by ( d D α )( X , X , . . . , X k ) = k (cid:88) i =0 ( − i ρ D ( X i )( α ( X , . . . , ˆ X i , . . . , X k ))+ (cid:88) i Let τ D : D → Q be a vector bundle over a manifold Q and D ∗ be its dual vectorbundle. Then, there exists a one-to-one correspondence between the set LAP ( D ∗ ) of linear almostPoisson structures on D ∗ , the set SSA ( D ) of skew-symmetric algebroid structures on τ D : D → Q and the set AD ( D ) of almost differentials on this vector bundle. AMILTON-JACOBI EQUATION AND NONHOLONOMIC MECHANICS 9 Skew-symmetric algebroids and the orbit theorem Let ( D, [[ · , · ]] D , ρ D ) be a skew-symmetric algebroid over Q and d D be the corresponding almostdifferential.We can consider the vector space over R H ( d D ) = { f ∈ C ∞ ( Q ) /d D f = 0 } . Note that if D is a Lie algebroid we have that H ( d D ) is the 0 Lie algebroid cohomology groupassociated with D .On the other hand, it is clear that if Q is connected and D is a transitive skew-symmetric algebroid,that is, ρ D ( D q ) = T q Q, for all q ∈ Q then H ( d D ) (cid:39) R . (3.1)Condition (3.1) will play an important role in Section 4.1.Next, we will see that (3.1) holds if the skew-symmetric algebroid is completely nonholonomic withconnected base space.Let ˜ D be the generalized distribution on Q whose characteristic space at the point q ∈ Q is˜ D q = ρ D ( D q ) . It is clear that ˜ D is finitely generated. Note that the C ∞ -module Γ( D ) is finitely generated (see[17]).Now, denote by Lie ∞ ( ˜ D ) the smallest Lie subalgebra of X ( Q ) containing ˜ D . Then Lie ∞ ( ˜ D ) iscomprised of finite R -linear combinations of vector fields of the form[ ˜ X k , [ ˜ X k − , . . . [ ˜ X , ˜ X ] . . . ]]with k ∈ N , k (cid:54) = 0, and ˜ X , . . . , ˜ X k ∈ X ( Q ) satisfying˜ X l ( q ) ∈ ˜ D q , for all q ∈ Q (see [2]).For each q ∈ Q , we will consider the vector subspace Lie ∞ q ( ˜ D ) of T q Q given byLie ∞ q ( ˜ D ) = { ˜ X ( q ) ∈ T q Q/ ˜ X ∈ Lie ∞ ( ˜ D ) } . Then, the assignment q ∈ Q → Lie ∞ q ( ˜ D ) ⊆ T q Q defines a generalized foliation on Q . The leaf L of this foliation over the point q ∈ Q is the orbitof ˜ D over the point q , that is, L = { ( φ ˜ X k ˜ t k ◦ · · · ◦ φ ˜ X ˜ t )( q ) ∈ Q/ ˜ t l ∈ R , ˜ X l ∈ X ( Q ) and ˜ X l ( q ) ∈ ˜ D q , for all q ∈ Q } . Here, φ ˜ X l ˜ t l is the flow of the vector field ˜ X l at the time ˜ t l (for more details, see [2]). Definition 3.1. The skew-symmetric algebroid ( D, [[ · , · ]] D , ρ D ) over Q is said to be completelynonholonomic if Lie ∞ q ( ρ D ( D )) = Lie ∞ q ( ˜ D ) = T q Q, for all q ∈ Q. Thus, if Q is a connected manifold, it follows that D is completely nonholonomic if and only if theorbit of ˜ D over any point q ∈ Q is Q . Remark 3.2. (i) Definition 3.1 may be extended for anchored vector bundles. A vector bundle τ D : D → Q over Q is said to be anchored if it admits an anchor map, that is, a vectorbundle morphism ρ D : D → T Q . In such a case, the vector bundle is said to be completelynonholonomic if Lie ∞ q ( ρ D ( D )) = Lie ∞ q ( ˜ D ) = T q Q , for all q ∈ Q .(ii) If D is a regular distribution on a manifold Q then the inclusion map i D : D → T Q is an anchormap for the vector bundle τ D : D → Q . Moreover, the anchored vector bundle τ D : D → Q iscompletely nonholonomic if and only if the distribution D is completely nonholonomic in theclassical sense of Vershik and Gershkovich [44]. In this sense it is formulated in the literaturethe classical Rashevsky-Chow theorem : If Lie ∞ q ( D ) = T q Q , for all q ∈ Q , then each orbitis equal to the whole manifold Q . (cid:5) Now, we deduce the following result Proposition 3.3. If the skew-symmetric algebroid ( D, [[ · , · ]] D , ρ D ) over Q is completely nonholonomicand Q is connected then H ( d D ) (cid:39) R .Proof. Suppose that f ∈ C ∞ ( Q ) and that d D f = 0. Let q be a point of Q . We must prove that˜ v ( f ) = 0 , for all ˜ v ∈ T q Q. The condition ( d D f )( q ) = 0 implies that ˜ v ( f ) = 0, for all ˜ v ∈ ˜ D q .Thus, first we have that 0 = ˜ X ( ˜ X ( f )) − ˜ X ( ˜ X ( f )) = [ ˜ X , ˜ X ]( f ) , for ˜ X , ˜ X ∈ X ( Q ) and ˜ X ( q ) , ˜ X ( q ) ∈ ˜ D q , for all q ∈ Q .Secondly, since D is completely nonholonomic then Lie ∞ q ( ˜ D ) = T q Q . Therefore, there exists afinite sequence of vector fields on Q , ˜ X , . . . , ˜ X k such that ˜ X i ( q ) ∈ ˜ D q , for all i ∈ { , . . . , k } and q ∈ Q , ˜ v = [ ˜ X k , [ ˜ X k − , . . . [ ˜ X , ˜ X ] . . . ]]( q ) . From both considerations, we deduce the result. (cid:3) However, the condition H ( d D ) (cid:39) R does not imply, in general, that the skew-symmetric algebroid D is completely nonholonomic.In fact, let D be the tangent bundle to R τ T R : T R → R . If ( x, y ) are the standard coordinates on R , it follows that { X = ∂∂x , X = ∂∂y } is a global basisof Γ( T R ) = X ( R ). So, we can consider the skew-symmetric algebroid structure ([[ · , · ]] T R , ρ T R ) on T R which is characterized by the following conditions[[ X , X ]] T R = 0 , ρ T R ( X ) = ∂∂x , ρ T R ( X ) = xy ∂∂y . AMILTON-JACOBI EQUATION AND NONHOLONOMIC MECHANICS 11 Then, the generalized distribution ˜ D = (cid:103) T R on R is generated by the vector fields˜ X = ∂∂x , ˜ X = xy ∂∂y . Thus, the Lie subalgebra Lie ∞ ( ˜ D ) of T R is generated by the vector fields˜ X = ∂∂x , ˜ X = xy ∂∂y , ˜ X = y ∂∂y . This implies that Lie ∞ ( x ,y ) ( ˜ D ) (cid:54) = T ( x ,y ) R , if y = 0 . However, if f ∈ C ∞ ( R ) and d D f = 0, we deduce that ∂f∂x = 0 , xy ∂f∂y = 0 . Consequently, using that f ∈ C ∞ ( R ), we obtain that f is constant.Next, we will discuss the case when the generalized foliation Lie ∞ ( ˜ D ) (cid:54) = T Q . In fact, we will provethe following result. Theorem 3.4. Let ( D, [[ · , · ]] D , ρ D ) be a skew-symmetric algebroid over a manifold Q and f be a real C ∞ -function on Q such that d D f = 0 . Suppose that L is an orbit of ˜ D and that D L is the vectorbundle over L given by D L = ∪ q ∈ L D q = τ − D ( L ) . Then: (i) The couple ([[ · , · ]] D , ρ D ) induces a skew-symmetric algebroid structure ([[ · , · ]] D L , ρ D L ) on thevector bundle τ D L : D L → L and the skew-symmetric algebroid ( D L , [[ · , · ]] D L , ρ D L ) is completelynonholonomic. (ii) The restriction of f to L is constant.Proof. It is clear that ρ D ( D q ) = ˜ D q ⊆ Lie ∞ q ( ˜ D ) = T q L, for all q ∈ L. Thus, we have a vector bundle morphism ρ D L : D L → T L. More precisely, ρ D L = ( ρ D ) | D L .On the other hand, we may define a R -bilinear skew-symmetric bracket[[ · , · ]] D L : Γ( D L ) × Γ( D L ) → Γ( D L ) . In fact, if X L , Y L ∈ Γ( D L ) then[[ X L , Y L ]] D L ( q ) = [[ X, Y ]] D ( q ) , for all q ∈ L where X, Y are sections of τ D : D → Q such that X | U ∩ L = ( X L ) | U ∩ L , Y | U ∩ L = ( Y L ) | U ∩ L , with U an open subset of Q and q ∈ U .Note that the condition Lie ∞ q ( ˜ D ) = T q L, for all q ∈ L, implies that ([[ X, Y ]] D ) | V ∩ L = 0 for X, Y ∈ Γ( D ), with V an open subset of Q and X ( q ) = 0 , for all q ∈ V ∩ L. Therefore, the map [[ · , · ]] D L is well-defined. We remark that if q ∈ V ∩ L and { X α } is a local basisof Γ( τ D ) in an open susbset W of Q , with q ∈ W , such that X = f α X α in W then, using that( f α ) | V ∩ W ∩ L = 0 and that ρ D ( Y )( q ) ∈ T q L , we deduce that[[ X, Y ]] D ( q ) = f α ( q )[[ X α , Y ]] D ( q ) − ρ D ( Y )( q )( f α ) X α ( q ) = 0 . Moreover, the couple ([[ · , · ]] D L , ρ D L ) is a skew-symmetric algebroid structure on the vector bundle τ D L : D L → L .In addition, it is clear that( (cid:102) D L ) q = ρ D L (( D L ) q ) = ˜ D q , Lie ∞ q ( (cid:102) D L ) = Lie ∞ q ( ˜ D )for all q ∈ L . Thus, we have that the skew-symmetric algebroid ( D L , [[ · , · ]] D L , ρ D L ) is completelynonholonomic.This proves (i).On the other hand, it follows that the condition d D f = 0 implies that d D L ( f | L ) = 0and, since H ( d D L ) (cid:39) R (as a consequence of the first part of the theorem), we conclude that f isconstant on L . (cid:3) It will be also interesting to characterize under what conditions there exist functions f ∈ C ∞ ( Q )such that ( d D ) f = 0. Using Equations (2.4) we easily deduce that: (cid:0) ( d D ) f (cid:1) ( X, Y ) = ([ ρ D ( X ) , ρ D ( Y )] − ρ D [[ X, Y ]] D ) f for all X, Y ∈ Γ( τ D ). Now, consider the generalized distribution ¯ D on Q whose characteristic space¯ D q at the point q ∈ Q is:¯ D q = { ([ ρ D ( X ) , ρ D ( Y )] − ρ D [[ X, Y ]] D )( q ) / X, Y ∈ Γ( τ D ) } . It is also clear that ¯ D is finitely generated. Denote by Lie ∞ ( ¯ D ) the smallest Lie subalgebra of X ( Q )containing ¯ D . Observe that Lie ∞ ( ¯ D ) ⊆ Lie ∞ ( ˜ D ). We deduce that ( d D ) f = 0 if and only if f isconstant on any orbit ¯ L of ¯ D . Of course if ¯ D is completely nonholonomic then the unique functions f ∈ C ∞ ( Q ) satisfying ( d D ) f = 0 are f = constant, but it has not to be always the case and it mayuseful to find this particular type of functions in concrete examples.4. Linear almost Poisson structures and Hamilton-Jacobi equation Linear almost Poisson structures and Hamiltonian systems. In this section, we willconsider Hamiltonian systems associated with a linear almost Poisson structure on the dual bundle D ∗ to a vector bundle and with a Hamiltonian function on D ∗ . Thus, the ingredients of our theoryare: (i) A vector bundle τ D : D → Q of rank n over a manifold Q of dimension m ;(ii) A linear almost Poisson structure {· , ·} D ∗ on D ∗ and(iii) A Hamiltonian function h : D ∗ → R on D ∗ . AMILTON-JACOBI EQUATION AND NONHOLONOMIC MECHANICS 13 The triplet ( D, {· , ·} D ∗ , h ) is said to be a Hamiltonian system .We will denote by Λ D ∗ the linear almost Poisson 2-vector associated with {· , ·} D ∗ . Then, we mayintroduce the vector field H Λ D ∗ h on D ∗ given by H Λ D ∗ h = − i ( dh )Λ D ∗ . H Λ D ∗ h is called the Hamiltonian vector field of h with respect to Λ D ∗ . The integral curves of H Λ D ∗ h are the solutions of the Hamilton equations for h .Now, suppose that ( q i ) are local coordinates on an open subset U of Q and that { X α } is a basisof the space of sections of the vector bundle τ − D ( U ) → U . Denote by ( q i , p α ) the corresponding localcoordinates on D ∗ and by C γαβ and ρ iα the local structure functions (with respect to the coordinates( q j ) and to the basis { X α } ) of the corresponding skew-symmetric algebroid structure on D . Then,using (2.1), it follows that H Λ D ∗ h = ρ iα ∂h∂p α ∂∂q i − ( ρ iα ∂h∂q i + C γαβ p γ ∂h∂p β ) ∂∂p α . (4.1)Therefore, the Hamilton equations are dq i dt = ρ iα ∂h∂p α , dp α dt = − ( ρ iα ∂h∂q i + C γαβ p γ ∂h∂p β ) . Hamiltonian systems and Hamilton-Jacobi equation. Let ( D, {· , ·} D ∗ , h ) be a Hamilton-ian system and α : Q → D ∗ be a section of the vector bundle τ D ∗ : D ∗ → Q .If H Λ D ∗ h is the Hamiltonian vector field of h with respect to {· , ·} D ∗ , we may introduce the vectorfield H Λ D ∗ h,α on Q given by H Λ D ∗ h,α ( q ) = ( T α ( q ) τ D ∗ )( H Λ D ∗ h ( α ( q ))) , for q ∈ Q. From (4.1), it follows that H Λ D ∗ h,α ( q ) ∈ ρ D ( D q ) , for all q ∈ Q, (4.2)where ([[ · , · ]] D , ρ D ) is the induced skew-symmetric algebroid structure on the vector bundle τ D : D → Q .Then, the aim of this section is to prove the following result. Theorem 4.1. Let ( D, {· , ·} D ∗ , h ) be a Hamiltonian system and α : Q → D ∗ be a section of thevector bundle τ D ∗ : D ∗ → Q such that d D α = 0 . Under these hypotheses, the following conditions areequivalent: (i) If c : I → Q is an integral curve of the vector field H Λ D ∗ h,α , that is, ˙ c ( t ) = ( T α ( c ( t )) τ D ∗ )( H Λ D ∗ h ( α ( c ( t )))) , for all t ∈ I, then α ◦ c : I → D ∗ is a solution of the Hamilton equations for h . (ii) α satisfies the Hamilton-Jacobi equation d D ( h ◦ α ) = 0 . Remark 4.2. Let ˜ D be the generalized distribution on Q given by ˜ D = ρ D ( D ) and Lie ∞ ( ˜ D ) bethe smallest Lie subalgebra of X ( Q ) containing ˜ D . Then, using Theorem 3.4, we deduce that theHamilton-Jacobi equation holds for the section α if and only if the function h ◦ α : Q → R is constanton the leaves of the foliation Lie ∞ ( ˜ D ). (cid:5) In order to prove Theorem 4.1, we will need some previous results: Proposition 4.3. Let {· , ·} D ∗ be a linear almost Poisson structure on the dual bundle D ∗ to a vectorbundle τ D : D → Q , α : Q → D ∗ be a section of τ D ∗ : D ∗ → Q and Λ D ∗ : T ∗ D ∗ → T D ∗ be thevector bundle morphism between T ∗ D ∗ and T D ∗ induced by the linear almost Poisson -vector Λ D ∗ .Then, α is a -cocycle with respect to d D (i.e., d D α = 0 ) if and only if for every point q of Q thesubspace of T α ( q ) D ∗ L α,D ( q ) = ( T q α )( ρ D ( D q )) (4.3) is Lagrangian with respect to Λ D ∗ , that is, Λ D ∗ (( L α,D ( q )) ) = L α,D ( q ) , for all q ∈ Q. Remark 4.4. If D = T Q and {· , ·} T ∗ Q is the canonical Poisson (symplectic) structure on T ∗ Q then α is a 1-form on Q , d D = d T Q is the standard exterior differential on Q and ρ D = ρ T Q : T Q → T Q isthe identity map. Thus, if we apply Proposition 4.3 we obtain that α is a closed 1-form if and onlyif α ( Q ) is a Lagrangian submanifold of T ∗ Q . This is a well-known result in the literature (see, forinstance, [1]). (cid:5) Proposition 4.5. Under the same hypotheses as in Proposition 4.3, if the section α is a -cocyclewith respect to d D , i.e., d D α = 0 then we have that Ker ( Λ D ∗ ( α ( q ))) ⊆ ( L α,D ( q )) , for all q ∈ Q. The proofs of Propositions 4.3 and 4.5 may be found in the Appendix of this paper. Proof of Theorem 4.1. It is clear that condition (i) in Theorem 4.1 is equivalent to the following fact:(i’) The vector fields H Λ D ∗ h,α and H Λ D ∗ h on Q and D ∗ , respectively, are α -related, that is, ( T q α )( H Λ D ∗ h,α ( q )) = H Λ D ∗ h ( α ( q )) , f or all q ∈ Q. (4.4)Therefore, we must prove that (i’) ⇐⇒ (ii)(i’) = ⇒ (ii) Let q be a point of Q . Then, using (4.2), (4.3) and (4.4), we deduce that H Λ D ∗ h ( α ( q )) ∈ L α,D ( q ) . Consequently, from Proposition 4.3, we obtain that H Λ D ∗ h ( α ( q )) = Λ D ∗ ( η α ( q ) ) , for some η α ( q ) ∈ ( L α,D ( q )) . Thus, since H Λ D ∗ h ( α ( q )) = − Λ D ∗ ( dh ( α ( q ))), it follows that η α ( q ) + dh ( α ( q )) ∈ Ker ( Λ D ∗ ( α ( q ))) . Now, using Proposition 4.5 and the fact that η α ( q ) ∈ ( L α,D ( q )) , we conclude that dh ( α ( q )) ∈ ( L α,D ( q )) . (4.5) AMILTON-JACOBI EQUATION AND NONHOLONOMIC MECHANICS 15 Finally, if a q ∈ D q , we have that d D ( h ◦ α )( q )( a q ) = dh ( α ( q ))(( T q α )( ρ D ( a q )))which implies that (see (4.3) and (4.5)) d D ( h ◦ α )( q )( a q ) = 0 . (ii) = ⇒ (i’) Let q be a point of Q . Then, using that d D ( h ◦ α )( q ) = 0, we deduce that dh ( α ( q )) ∈ ( L α,D ( q )) . Therefore, it follows that H Λ D ∗ h ( α ( q )) = − Λ D ∗ ( dh ( α ( q ))) ∈ Λ D ∗ (( L α,D ( q )) )and, from Proposition 4.3, we obtain that there exists v q ∈ ρ D ( D q ) ⊆ T q Q such that H Λ D ∗ h ( α ( q )) = ( T q α )( v q ) . This implies that H Λ D ∗ h,α ( q ) = ( T α ( q ) τ D ∗ )( H Λ D ∗ h ( α ( q )) = v q and, thus, H Λ D ∗ h ( α ( q )) = ( T q α )( H Λ D ∗ h,α ( q )) . (cid:3) Let ( D, {· , ·} D ∗ , h ) be a Hamiltonian system and α : Q → D ∗ be a section of the vector bundle τ D ∗ : D ∗ → Q .Suppose that ( q i ) are local coordinates on an open subset U of Q and that { X γ } is a basis ofsections of the vector bundle τ − D ( U ) → U . Denote by ( q i , p γ ) the corresponding local coordinates on D ∗ and by ρ iγ , C δγν the local structure functions of the skew-symmetric algebroid structure ([[ · , · ]] D , ρ D )with respect to the local coordinates ( q i ) and to the basis { X γ } . If the local expression of α is α ( q i ) = ( q i , α γ ( q i ))then d D α = 0 ⇐⇒ C δγν α δ = ρ iγ ∂α ν ∂q i − ρ iν ∂α γ ∂q i , ∀ γ, ν, and d D ( h ◦ α ) = 0 ⇐⇒ ρ iγ ( q )( ∂h∂q i | α ( q ) + ∂α ν ∂q i | q ∂h∂p ν | α ( q ) ) = 0 , ∀ γ, ∀ q ∈ U. Corollary 4.6. Under the same hypotheses as in Theorem 4.1 if, additionally, H ( d D ) (cid:39) R or ifthe skew-symmetric algebroid ( D, [[ · , ]] D , ρ D ) is completely nonholonomic and Q is connected, then thefollowing conditions are equivalent: (i) If c : I → Q is an integral curve of the vector field H Λ D ∗ h,α , that is, ˙ c ( t ) = ( T α ( c ( t )) τ D ∗ )( H Λ D ∗ h ( α ( c ( t )))) , for all t ∈ I, then α ◦ c : I → D ∗ is a solution of the Hamilton equations for h . (ii) α satisfies the following relation h ◦ α = constant . Note that if Q is connected then h ◦ α = constant ⇐⇒ ( ∂h∂q i | α ( q ) + ∂α ν ∂q i | q ∂h∂p ν | α ( q ) ) = 0 , ∀ i and ∀ q ∈ U. Linear almost Poisson morphisms and Hamilton-Jacobi equation. Suppose that τ D : D → Q and τ ¯ D : ¯ D → ¯ Q are vector bundles over Q and ¯ Q , respectively, and that {· , ·} D ∗ (respectively, {· , ·} ¯ D ∗ ) is a linear almost Poisson structure on D ∗ (respectively, ¯ D ∗ ). Denote by ([[ · , · ]] D , ρ D ) and d D (respectively, ([[ · , · ]] ¯ D , ρ ¯ D ) and d ¯ D ) the corresponding skew-symmetric algebroid structure and thealmost differential on the vector bundle τ D : D → Q (respectively, τ ¯ D : ¯ D → ¯ Q ). Definition 4.7. A vector bundle morphism ( ˜ F , F ) between the vector bundles τ D ∗ : D ∗ → Q and τ ¯ D ∗ : ¯ D ∗ → ¯ Q Q F (cid:45) ¯ Qτ D ∗ (cid:63) τ ¯ D ∗ (cid:63) D ∗ ˜ F (cid:45) ¯ D ∗ is said to be a linear almost Poisson morphism if { ¯ ϕ ◦ ˜ F , ¯ ψ ◦ ˜ F } D ∗ = { ¯ ϕ, ¯ ψ } ¯ D ∗ ◦ ˜ F , (4.6) for ¯ ϕ, ¯ ψ ∈ C ∞ ( ¯ D ∗ ) . Let ( ˜ F , F ) be a vector bundle morphism between the vector bundles τ D ∗ : D ∗ → Q and τ ¯ D ∗ :¯ D ∗ → ¯ Q . If ¯ X is a section of τ ¯ D : ¯ D → ¯ Q then we may define the section ( ˜ F , F ) ∗ ¯ X of τ D : D → Q characterized by the following condition α q ((( ˜ F , F ) ∗ ¯ X )( q )) = ˜ F ( α q )( ¯ X ( F ( q ))) , (4.7)for all q ∈ Q and α q ∈ D ∗ q . Theorem 4.8. Let ( ˜ F , F ) be a vector bundle morphism between the vector bundles τ D ∗ : D ∗ → Q and τ ¯ D ∗ : ¯ D ∗ → ¯ Q . Then, ( ˜ F , F ) is a linear almost Poisson morphism if and only if [[( ˜ F , F ) ∗ ¯ X, ( ˜ F , F ) ∗ ¯ Y ]] D = ( ˜ F , F ) ∗ [[ ¯ X, ¯ Y ]] ¯ D , (4.8)( T F ◦ ρ D )(( ˜ F , F ) ∗ ¯ X ) = ρ ¯ D ( ¯ X ) ◦ F, (4.9) for ¯ X, ¯ Y ∈ Γ( τ ¯ D ) .Proof. Suppose that ( ˜ F , F ) is a linear almost Poisson morphism and that ¯ Z is a section of τ ¯ D : ¯ D → ¯ Q . From (4.7), it follows that ( ˜ F , F ) ∗ ¯ Z = ˆ¯ Z ◦ ˜ F . (4.10)Now, if ¯ X, ¯ Y ∈ Γ( τ D ) then, using (2.2) and (4.10), we deduce that[[( ˜ F , F ) ∗ ¯ X, ( ˜ F , F ) ∗ ¯ Y ]] D = −{ ˆ¯ X ◦ ˜ F , ˆ¯ Y ◦ ˜ F } D ∗ . AMILTON-JACOBI EQUATION AND NONHOLONOMIC MECHANICS 17 Thus, from (2.2) and (4.6), we obtain that[[( ˜ F , F ) ∗ ¯ X, ( ˜ F , F ) ∗ ¯ Y ]] D = ( ˜ F , F ) ∗ [[ ¯ X, ¯ Y ]] ¯ D which implies that (4.8) holds.On the other hand, if ¯ f ∈ C ∞ ( ¯ Q ) then, using again (2.2) and (4.6), it follows that( ρ ¯ D ( ¯ X )( ¯ f ) ◦ F ) ◦ τ D ∗ = { ( ¯ f ◦ τ ¯ D ∗ ) ◦ ˜ F , ˆ¯ X ◦ ˜ F } D ∗ . Therefore, from (2.2) and (4.10), we have that( ρ ¯ D ( ¯ X )( ¯ f ) ◦ F ) ◦ τ D ∗ = ρ D (( ˜ F , F ) ∗ ¯ X )( ¯ f ◦ F ) ◦ τ D ∗ and, consequently, ρ ¯ D ( ¯ X )( ¯ f ) ◦ F = ρ D (( ˜ F , F ) ∗ ¯ X )( ¯ f ◦ F ) . This implies that (4.9) holds.Conversely, assume that (4.8) and (4.9) hold.Then, if ¯ f , ¯ g ∈ C ∞ ( ¯ Q ) it is clear that the real functions¯ f ◦ τ ¯ D ∗ ◦ ˜ F = ¯ f ◦ F ◦ τ D ∗ , ¯ g ◦ τ ¯ D ∗ ◦ ˜ F = ¯ g ◦ F ◦ τ D ∗ are basic functions with respect to the projection τ D ∗ : D ∗ → Q . Therefore, we deduce that0 = { ( ¯ f ◦ τ ¯ D ∗ ) ◦ ˜ F , (¯ g ◦ τ ¯ D ∗ ) ◦ ˜ F } D ∗ = { ¯ f ◦ τ ¯ D ∗ , ¯ g ◦ τ ¯ D ∗ } ¯ D ∗ ◦ ˜ F . (4.11)Now, if ¯ X ∈ Γ( τ ¯ D ) then, using (2.2) and (4.10), we obtain that { ( ¯ f ◦ τ ¯ D ∗ ) ◦ ˜ F , ˆ¯ X ◦ ˜ F } D ∗ = (( T F ◦ ρ D )(( ˜ F , F ) ∗ ¯ X ))( ¯ f ) ◦ τ D ∗ . Consequently, from (2.2) and (4.9), it follows that { ( ¯ f ◦ τ ¯ D ∗ ) ◦ ˜ F , ˆ¯ X ◦ ˜ F } D ∗ = { ¯ f ◦ τ ¯ D ∗ , ˆ¯ X } ¯ D ∗ ◦ ˜ F . (4.12)On the other hand, if ¯ Y ∈ Γ( τ ¯ D ) then, using (2.2), (4.8) and (4.10), we deduce that { ˆ¯ X ◦ ˜ F , ˆ¯ Y ◦ ˜ F } D ∗ = { ˆ¯ X, ˆ¯ Y } ¯ D ∗ ◦ ˜ F . (4.13)Thus, (4.11), (4.12) and (4.13) imply that ( ˜ F , F ) is a linear almost Poisson morphism. (cid:3) Let ( ˜ F , F ) be a vector bundle morphism between the vector bundles τ D ∗ : D ∗ → Q and τ ¯ D ∗ :¯ D ∗ → ¯ Q . Denote by Λ k ˜ F : Λ k D ∗ → Λ k ¯ D ∗ the vector bundle morphism (over F ) between the vectorbundles Λ k τ D ∗ : Λ k D ∗ → Q and Λ k τ ¯ D ∗ : Λ k ¯ D ∗ → ¯ Q induced by ˜ F . Then, a section α ∈ Γ(Λ k τ D ∗ ) issaid to be ( ˜ F , F )-related with a section ¯ α ∈ Γ(Λ k τ ¯ D ∗ ) ifΛ k ˜ F ◦ α = ¯ α ◦ F. Now, assume that F is a surjective map and that ( ˜ F , F ) is a fiberwise injective vector bundlemorphism, that is, ˜ F q = ˜ F | D ∗ q : D ∗ q → ¯ D ∗ F ( q ) is a monomorphism of vector spaces, for all q ∈ Q , and F ( q ) = F ( q (cid:48) ) = ⇒ ˜ F q ( D ∗ q ) = ˜ F q (cid:48) ( D ∗ q (cid:48) ) . Then, we may consider the vector subbundle ˜ F ( D ∗ ) (over ¯ Q ) of τ ¯ D ∗ : ¯ D ∗ → ¯ Q . Moreover, if ¯ α is asection of this vector subbundle we have that there exists a unique section α of τ D ∗ : D ∗ → Q such that α is ( ˜ F , F )-related with ¯ α . In fact, if { ¯ α i } is a local basis of sections of τ ˜ F ( D ∗ ) : ˜ F ( D ∗ ) → ¯ Q , itfollows that { α i } is a local basis of Γ( τ D ∗ ). Theorem 4.9. Let ( ˜ F , F ) be a vector bundle morphism between the vector bundles τ D ∗ : D ∗ → Q and τ ¯ D ∗ : ¯ D ∗ → ¯ Q . (i) If ( ˜ F , F ) is a linear almost Poisson morphism then the following condition ( C ) holds:( C ) For each α ∈ Γ(Λ k τ D ∗ ) which is ( ˜ F , F ) -related with ¯ α ∈ Γ(Λ k τ ¯ D ∗ ) we have that d D α ∈ Γ(Λ k +1 τ D ∗ ) is also ( ˜ F , F ) -related with d ¯ D ¯ α ∈ Γ(Λ k +1 τ ¯ D ∗ ) . (ii) Conversely, if condition ( C ) holds, F is a surjective map and ( ˜ F , F ) is a fiberwise injectivevector bundle morphism then ( ˜ F , F ) is a linear almost Poisson morphism.Proof. (i) Suppose that ¯ X and ¯ Y are sections of τ ¯ D : ¯ D → ¯ Q .Then, if ¯ f ∈ C ∞ ( ¯ Q ), using (4.9), we deduce that( d ¯ D ¯ f )( ¯ X ) ◦ F = ( ρ D (( ˜ F , F ) ∗ ¯ X ))( ¯ f ◦ F ) . Thus, from (4.7), it follows that( d ¯ D ¯ f )( ¯ X ) ◦ F = < ˜ F ( d D ( ¯ f ◦ F )) , ¯ X ◦ F > . Therefore, we have that d ¯ D ¯ f ◦ F = ˜ F ◦ d D ( ¯ f ◦ F ) . (4.14)Now, let α be a section of τ D ∗ : D ∗ → Q which is ( ˜ F , F )-related with ¯ α ∈ Γ( τ ¯ D ∗ ), that is,˜ F ◦ α = ¯ α ◦ F. (4.15)Then, using (4.8), (4.9) and (4.15), we obtain that( d ¯ D ¯ α )( ¯ X, ¯ Y ) ◦ F = ρ D (( ˜ F , F ) ∗ ¯ X )( α (( ˜ F , F ) ∗ ¯ Y )) − ρ D (( ˜ F , F ) ∗ ¯ Y )( α (( ˜ F , F ) ∗ ¯ X )) − α [[( ˜ F , F ) ∗ ¯ X, ( ˜ F , F ) ∗ ¯ Y ]] D which implies that ( d ¯ D ¯ α )( ¯ X, ¯ Y ) ◦ F = < Λ ˜ F ◦ d D α, ( ¯ X ◦ F, ¯ Y ◦ F ) > . This proves that d ¯ D ¯ α ◦ F = Λ ˜ F ◦ d D α. (4.16)Consequently, from (2.3), (4.14) and (4.16), we deduce the result.(ii) If ¯ X ∈ Γ( τ ¯ D ) and ¯ f ∈ C ∞ ( ¯ Q ) then, using condition ( C ), we have that( ρ ¯ D ( ¯ X ) ◦ F )( ¯ f ) = d D ( ¯ f ◦ F )(( ˜ F , F ) ∗ ¯ X ) = ( ρ D (( ˜ F , F ) ∗ ¯ X ))( ¯ f ◦ F ) . This proves that (4.9) holds.Next, suppose that ¯ Y ∈ Γ( τ ¯ D ) and that α is a section of τ D ∗ : D ∗ → Q which is ( ˜ F , F )-relatedwith ¯ α ∈ Γ( τ ¯ D ∗ ).Then, from (4.9), it follows that α [[( ˜ F , F ) ∗ ¯ X, ( ˜ F , F ) ∗ ¯ Y ]] D = − ( d D α )(( ˜ F , F ) ∗ ¯ X, ( ˜ F , F ) ∗ ¯ Y ) + ρ ¯ D ( ¯ X )( ¯ α ( ¯ Y )) ◦ F − ρ ¯ D ( ¯ Y )( ¯ α ( ¯ X )) ◦ F. Thus, using condition ( C ), we deduce that α [[( ˜ F , F ) ∗ ¯ X, ( ˜ F , F ) ∗ ¯ Y ]] D = − ( d ¯ D ¯ α )( ¯ X, ¯ Y ) ◦ F + ρ ¯ D ( ¯ X )( ¯ α ( ¯ Y )) ◦ F − ρ ¯ D ( ¯ Y )( ¯ α ( ¯ X )) ◦ F. AMILTON-JACOBI EQUATION AND NONHOLONOMIC MECHANICS 19 This implies that (4.8) holds. (cid:3) Remark 4.10. Let ( ˜ F , F ) be a linear almost Poisson morphism between the vector bundles τ D ∗ : D ∗ → Q and τ ¯ D ∗ : ¯ D ∗ → ¯ Q . Moreover, suppose that F is surjective and that ( ˜ F , F ) is a fiberwiseinjective vector bundle morphism.(i) From Theorem 4.9, we deduce that the condition H ( d D ) (cid:39) R implies that H ( d ¯ D ) (cid:39) R . Ingeneral, the converse does not hold. However, if H ( d ¯ D ) (cid:39) R and f ∈ C ∞ ( Q ) is a F -basicfunction such that d D f = 0 then f is constant.(ii) If F is a surjective submersion with connected fibers, V q F ⊆ ˜ D q = ρ D ( D q ), for all q ∈ Q , and d D f = 0 then f is a F -basic function. Here, V F is the vertical bundle to F . (cid:5) Now, we will introduce the following definition. Definition 4.11. Let ( D, {· , ·} D ∗ , h ) (respectively, ( ¯ D, {· , ·} ¯ D ∗ , ¯ h ) ) be a Hamiltonian system and ( ˜ F , F ) be a linear almost Poisson morphism between the vector bundles τ D : D ∗ → Q and τ ¯ D :¯ D ∗ → ¯ Q . Then, ( ˜ F , F ) is said to be Hamiltonian if ¯ h ◦ ˜ F = h. It is clear that if ( ˜ F , F ) is a Hamiltonian morphism then the Hamiltonian vector fields of h and¯ h , H Λ D ∗ h and H Λ ¯ D ∗ ¯ h , are ˜ F -related, that is,( T β ˜ F )( H Λ D ∗ h ( β )) = H Λ ¯ D ∗ ¯ h ( ˜ F ( β )) , for all β ∈ D ∗ . This implies that if µ : I → D ∗ is a solution of the Hamilton equations for h then ˜ F ◦ µ : I → ¯ D ∗ isa solution of the Hamilton equations for ¯ h .In addition, from Theorem 4.9, we deduce the following result Theorem 4.12. Let ( D, {· , ·} D ∗ , h ) (respectively, ( ¯ D, {· , ·} ¯ D ∗ , ¯ h ) ) be a Hamiltonian system and ( ˜ F , F ) be a Hamiltonian morphism between the vector bundles τ D ∗ : D ∗ → Q and τ ¯ D ∗ : ¯ D ∗ → ¯ Q . Assumethat the map F is surjective and that ( ˜ F , F ) is a fiberwise injective vector bundle morphism. (i) If α is a section of the vector bundle τ D ∗ : D ∗ → Q such that d D α = 0 , it satisfies theHamilton-Jacobi equation for h (respectively, the strongest condition h ◦ α = constant ) andit is ( ˜ F , F ) -related with ¯ α ∈ Γ( τ ¯ D ∗ ) then d ¯ D ¯ α = 0 and ¯ α satisfies the Hamilton-Jacobi equationfor ¯ h (respectively, the strongest condition ¯ h ◦ ¯ α = constant ). (ii) If ¯ α is a section of the vector subbundle ˜ F ( D ∗ ) of τ ¯ D ∗ : ¯ D ∗ → ¯ Q such that d ¯ D ¯ α = 0 and ¯ α satisfies the Hamilton-Jacobi equation for ¯ h (respectively, the strongest condition ¯ h ◦ ¯ α = constant ) then d D α = 0 and α satisfies the Hamilton-Jacobi equation for h (respectively, thestrongest condition h ◦ α = constant ), where α is the section of τ D ∗ : D ∗ → Q characterizedby the condition ˜ F ◦ α = ¯ α ◦ F . Applications to nonholonomic Mechanics Unconstrained mechanical systems on a Lie algebroid. Let τ A : A → Q be a Lie algebroidover a manifold Q and denote by ([[ · , · ]] A , ρ A ) the Lie algebroid structure of A . If G : A × Q A → R is a bundle metric on A then the Levi-Civita connection ∇ G : Γ( τ A ) × Γ( τ A ) → Γ( τ A )is determined by the formula2 G ( ∇ G X Y, Z ) = ρ A ( X )( G ( Y, Z )) + ρ A ( Y )( G ( X, Z )) − ρ A ( Z )( G ( X, Y ))+ G ( X, [[ Z, Y ]] A ) + G ( Y, [[ Z, X ]] A ) − G ( Z, [[ Y, X ]] A )for X, Y, Z ∈ Γ( A ). Using the covariant derivative induced by ∇ G , one may introduce the notion ofa geodesic of ∇ G as follows. A curve σ : I → A is admissible if ddt ( τ A ◦ σ ) = ρ A ◦ σ. An admissible curve σ : I → A is said to be a geodesic if ∇ G σ ( t ) σ ( t ) = 0, for all t ∈ I .The geodesics are the integral curves of a vector field ξ G on A , the geodesic flow of A , which islocally given by ξ G = ρ iB v B ∂∂q i − C CEB v B v C ∂∂v E . Here, ( q i ) are local coordinates on an open subset U of Q , { X B } is an orthonormal basis of sectionsof the vector bundle τ − A ( U ) → U , ( q i , v B ) are the corresponding local coordinates on A and ρ iB , C ECB are the local structure functions of A . Note that the coefficients Γ EBC of the connection ∇ G areΓ EBC = 12 ( C CEB + C BEC + C EBC )(for more details, see [7, 9]).The Lagrangian function L : A → R of an (unconstrained) mechanical system on A is definedby L ( a ) = 12 G ( a, a ) − V ( τ A ( a )) = 12 (cid:107) a (cid:107) G − V ( τ A ( a )) , for a ∈ A,V : Q → R being a real C ∞ -function on Q . In other words, L is the kinetic energy induced by G minus the potential energy induced by V .Note that if ∆ is the Liouville vector field of A then the Lagrangian energy E L = ∆( L ) − L isthe real C ∞ -function on A given by E L ( a ) = 12 G ( a, a ) + V ( τ A ( a )) = 12 (cid:107) a (cid:107) G + V ( τ A ( a )) , for a ∈ A. On the other hand, we may consider the section grad G V of τ A : A → Q characterized by the followingcondition G ( grad G V, X ) = ( d A V )( X ) = ρ A ( X )( V ) , ∀ X ∈ Γ( τ A ) . Then, the solutions of the Euler-Lagrange equations for L are the integral curves of the vectorfield ξ L on A defined by ξ L = ξ G − ( grad G V ) v , where ( grad G V ) v ∈ X ( A ) is the standard vertical lift of the section grad G V . The local expression ofthe Euler-Lagrange equations is˙ q i = ρ iB v B , ˙ v E = − C CEB v B v C − ρ jE ∂V∂q j , AMILTON-JACOBI EQUATION AND NONHOLONOMIC MECHANICS 21 for all i and E (see [7, 9]).Now, we will denote by (cid:91) G : A → A ∗ the vector bundle isomorphism induced by G and by G : A ∗ → A the inverse morphism. If α : Q → A ∗ is a section of the vector bundle τ A ∗ : A ∗ → Q we alsoconsider the vector field ξ L,α on Q defined by ξ L,α ( q ) = ( T G ( α ( q )) τ A )( ξ L ( G ( α ( q )))) , for q ∈ Q. Corollary 5.1. Let α : Q → A ∗ be a -cocycle of the Lie algebroid A , that is, d A α = 0 . Then, thefollowing conditions are equivalent: (i) If c : I → Q is an integral curve of the vector field ξ L,α on Q we have that G ◦ α ◦ c : I → A is a solution of the Euler-Lagrange equations for L . (ii) α satisfies the Hamilton-Jacobi equation d A ( E L ◦ G ◦ α ) = 0 , that is, the function (cid:107) G ◦ α (cid:107) G + V on Q is constant on the leaves of the Lie algebroidfoliation associated with A .Proof. The Legendre transformation associated with the Lagrangian function L is the vector bundleisomorphism (cid:91) G : A → A ∗ between A and A ∗ induced by the bundle metric G (for the definition of theLegendre transformation associated with a Lagrangian function on a Lie algebroid, see [25]). Thus,if we denote by G ∗ the bundle metric on A ∗ then, the Hamiltonian function H L = E L ◦ G inducedby the hyperregular Lagrangian function L is given by H L ( γ ) = 12 G ∗ ( γ, γ ) + V ( τ A ∗ ( γ )) , for γ ∈ A ∗ . Therefore, if Λ A ∗ is the corresponding linear Poisson 2-vector on A ∗ and H Λ A ∗ H L is the Hamiltonianvector field of H L with respect to Λ A ∗ , we have that the solutions of the Hamilton equations are theintegral curves of the vector field H Λ A ∗ H L . In fact, the vector fields ξ L and H Λ A ∗ H L are (cid:91) G -related, that is, T (cid:91) G ◦ ξ L = H Λ A ∗ H L ◦ (cid:91) G . Consequently, if σ : I → A is a solution of the Euler-Lagrange equations for L then (cid:91) G ◦ σ : I → A ∗ is a solution of the Hamilton equations for H L and, conversely, if γ : I → A ∗ is a solution of theHamilton equations for H L then G ◦ γ : I → A is a solution of the Euler-Lagrange equations for L (for more details, see [25]).In addition, since τ A ∗ ◦ (cid:91) G = τ A , it follows that ξ L,α ( q ) = ( T α ( q ) τ A ∗ )( H Λ A ∗ H L ( α ( q ))) = H Λ A ∗ H L ,α ( q ) , for q ∈ Q, i.e., ξ L,α = H Λ A ∗ H L ,α .Thus, using Theorem 4.1 (or, alternatively, using Theorem 3.16 in [25]), we deduce the result. (cid:3) Next, we will apply Corollary 5.1 to the particular case when A is the standard Lie algebroid T Q and α is a 1-coboundary, that is, α = dS with S : Q → R a real C ∞ -function on Q . Note that, inthis case, the bundle metric G on T Q is a Riemannian metric g on Q and that G ◦ α = g ◦ dS isjust the gradient vector field of S , grad g S , with respect to g . Corollary 5.2. Let S : Q → R be a real C ∞ -function on Q . Then, the following conditions areequivalent: (i) If c : I → Q is an integral curve of the vector field ξ L,dS on Q we have that grad g S ◦ c : I → A is a solution of the Euler-Lagrange equations for L . (ii) S satisfies the Hamilton-Jacobi equation d ( E L ◦ grad g S ) = 0 , that is, the function (cid:107) grad g S (cid:107) g + V on Q is constant. Remark 5.3. Corollary 5.2 is a consequence of a well-known result (see Theorem 5.2.4 in [1]). (cid:5) Now, let L : A → R (respectively, ¯ L : ¯ A → R ) be the Lagrangian function of an unconstrainedmechanical system on a Lie algebroid τ A : A → Q (respectively, τ ¯ A : ¯ A → ¯ Q ) and ( ˜ F , F ) be a linearPoisson morphism between the Poisson manifolds ( A ∗ , {· , ·} A ∗ ) and ( ¯ A ∗ , {· , ·} ¯ A ∗ ) such that:(i) F : Q → ¯ Q is a surjective map.(ii) For each q ∈ Q , the linear map ˜ F q = ˜ F | A ∗ q : A ∗ q → ¯ A ∗ F ( q ) satisfies the following conditions¯ G ∗ ( ˜ F q ( β ) , ˜ F q ( β (cid:48) )) = G ∗ ( β, β (cid:48) ) , for β, β (cid:48) ∈ A ∗ q ,F ( q ) = F ( q (cid:48) ) = ⇒ ˜ F q ( A ∗ q ) = ˜ F q (cid:48) ( A ∗ q (cid:48) ) , where G ∗ (respectively, ¯ G ∗ ) is the bundle metric on A ∗ (respectively, ¯ A ∗ ). Note that the firstcondition implies that ˜ F q is injective and an isometry.(iii) If V : Q → R (respectively, ¯ V : ¯ Q → R ) is the potential energy of the mechanical system on A (respectively, ¯ A ) we have that ¯ V ◦ F = V .Then, we deduce that ( ˜ F , F ) is a Hamiltonian morphism between the Hamiltonian systems ( A, {· , ·} A ∗ ,H L ) and ( ¯ A, {· , ·} ¯ A ∗ , H ¯ L ), where H L (respectively, H ¯ L ) is the Hamiltonian function on A ∗ (respec-tively, ¯ A ∗ ) associated with the Lagrangian function L (respectively, ¯ L ).Moreover, using Theorem 4.12, we conclude that Corollary 5.4. (i) If α : Q → A ∗ is a -cocycle for the Lie algebroid A ( d A α = 0 ), it satisfiesthe Hamilton-Jacobi equation d A ( E L ◦ G ◦ α ) = 0 (5.1) and it is ( ˜ F , F ) -related with ¯ α ∈ Γ( τ ¯ A ∗ ) then d ¯ A ¯ α = 0 and ¯ α is a solution of the Hamilton-Jacobi equation d ¯ A ( E ¯ L ◦ ¯ G ◦ ¯ α ) = 0 . (5.2)(ii) If ¯ α : ¯ Q → ˜ F ( A ∗ ) ⊆ ¯ A ∗ is a -cocycle for the Lie algebroid ¯ A ( d ¯ A ¯ α = 0 ) and it satisfiesthe Hamilton-Jacobi equation (5.2) then d A α = 0 and α is a solution of the Hamilton Jacobiequation (5.1). Here, α : Q → A ∗ is the section of τ A ∗ : A ∗ → Q characterized by thecondition ˜ F ◦ α = ¯ α ◦ F . A particular example of the above general construction is the following one.Let F : Q → ¯ Q = Q/G be a principal G -bundle. Denote by φ : G × Q → Q the free action of G on Q and by T φ : G × T Q → T Q the tangent lift of φ . T φ is a free action of G on T Q . Then, wemay consider the quotient vector bundle τ ¯ A = τ T Q/G : ¯ A = T Q/G → ¯ Q = Q/G . The sections of this AMILTON-JACOBI EQUATION AND NONHOLONOMIC MECHANICS 23 vector bundle may be identified with the vector fields on Q which are G -invariant. Thus, using that a G -invariant vector field is F -projectable and that the standard Lie bracket of two G -invariant vectorfields is also a G -invariant vector field, we can define a Lie algebroid structure ([[ · , · ]] ¯ A , ρ ¯ A ) on thequotient vector bundle τ ¯ A = τ T Q/G : ¯ A = T Q/G → ¯ Q = Q/G . The resultant Lie algebroid is calledthe Atiyah (gauge) algebroid associated with the principal bundle F : Q → ¯ Q = Q/G (see[25, 27]).On the other hand, denote by T ∗ φ : G × T ∗ Q → T ∗ Q the cotangent lift of the action φ . Then,the space of orbits of T ∗ φ , T ∗ Q/G , may be identified with the dual bundle ¯ A ∗ to ¯ A . Under thisidentification, the linear Poisson structure on ¯ A ∗ is characterized by the following condition: thecanonical projection ˜ F : A ∗ = T ∗ Q → T ∗ Q/G (cid:39) ¯ A ∗ is a Poisson morphism, when on A ∗ = T ∗ Q weconsider the linear Poisson structure induced by the standard Lie algebroid τ A = τ T Q : A = T Q → Q ,that is, the Poisson structure induced by the canonical symplectic structure of T ∗ Q (an explicitdescription of the linear Poisson structure on ¯ A ∗ (cid:39) T ∗ Q/G may be found in [32]).Thus, ( ˜ F , F ) is a linear Poisson morphism between A ∗ = T ∗ Q and ¯ A ∗ (cid:39) T ∗ Q/G and, in addition,˜ F is a fiberwise bijective vector bundle morphism.Now, suppose that G = g is a G -invariant Riemannian metric on Q and that V : Q → R is a G -invariant function on Q . Then, we may consider the corresponding mechanical Lagrangian function L : A = T Q → R . Moreover, it is clear that g and V induce a bundle metric ¯ G on ¯ A = T Q/G and areal function ¯ V : ¯ Q → R and, therefore, a mechanical Lagrangian function ¯ L : ¯ A = T Q/G → R .On the other hand, we have that for each q ∈ Q the map ˜ F q : A ∗ q = T ∗ q Q → ¯ A ∗ F ( q ) (cid:39) ( T ∗ Q/G ) F ( q ) is a linear isometry. Consequently, using Corollary 5.4, we deduce the following result Corollary 5.5. There exists a one-to-one correspondence between the -cocycles of the Atiyah alge-broid τ ¯ A = τ T Q/G : ¯ A = T Q/G → ¯ Q = Q/G which are solutions of the Hamilton-Jacobi equation forthe mechanical Lagrangian function ¯ L : ¯ A = T Q/G → R and the G -invariant closed -forms α on Q such that the function (cid:107) g ◦ α (cid:107) g + V is constant. An explicit example : The Elroy’s Beanie . This system is probably the most simple exampleof a dynamical system with a non-Abelian Lie group of symmetries. It consists in two planar rigidbodies attached at their centers of mass, moving freely in the plane (see [29]). So, the configurationspace is Q = SE (2) × S with coordinates q = ( x, y, θ, ψ ), where the three first coordinates describethe position and orientation of the center of mass of the first body and the last one the relativeorientation between both bodies. The Lagrangian L : T Q → R is L = 12 m ( ˙ x + ˙ y ) + 12 I ˙ θ + 12 I ( ˙ θ + ˙ ψ ) − V ( ψ )where m denotes the mass of the system and I and I are the inertias of the first and the secondbody, respectively; additionally, we also consider a potential function of the form V ( ψ ). The kineticenergy is associated with the Riemannian metric G on Q given by G = m ( dx + dy ) + ( I + I ) dθ + I dθ ⊗ dψ + I dψ ⊗ dθ + I dψ . The system is SE (2)-invariant for the actionΦ g ( q ) = ( z + x cos α − y sin α, z + x sin α + y cos α, α + θ, ψ )where g = ( z , z , α ). Let { ξ , ξ , ξ } be the standard basis of se (2),[ ξ , ξ ] = 0 , [ ξ , ξ ] = − ξ and [ ξ , ξ ] = ξ . The quotient space ¯ Q = Q/SE (2) = ( SE (2) × S ) /SE (2) (cid:39) S is naturally parameterized bythe coordinate ψ . The Atiyah algebroid T Q/SE (2) → ¯ Q is identified with the vector bundle: τ ¯ A : ¯ A = T S × s e (2) → S . The canonical basis of sections of τ ¯ A is: (cid:26) ∂∂ψ , ξ , ξ , ξ (cid:27) . Since themetric G is also SE (2)-invariant we obtain a bundle metric ¯ G and a ¯ G -orthonormal basis of sections: (cid:40) X = (cid:114) I + I I I (cid:18) ∂∂ψ − I I + I ξ (cid:19) , X = 1 √ m ξ , X = 1 √ m ξ , X = 1 √ I + I ξ (cid:41) In the coordinates ( ψ, v , v , v , v ) induced by the orthonormal basis of sections, the reduced La-grangian is ¯ L = 12 (cid:0) ( v ) + ( v ) + ( v ) + ( v ) (cid:1) − V ( ψ ) . Additionally, we deduce that[[ X , X ]] ¯ A = − (cid:115) I I ( I + I ) X , [[ X , X ]] ¯ A = (cid:115) I I ( I + I ) X , [[ X , X ]] ¯ A = 0 , [[ X , X ]] ¯ A = 0 , [[ X , X ]] ¯ A = − √ I + I X , [[ X , X ]] ¯ A = 1 √ I + I X . Therefore, the non-vanishing structure functions are C = − (cid:115) I I ( I + I ) , C = (cid:115) I I ( I + I ) , C = − √ I + I , C = 1 √ I + I . Moreover, ρ ¯ A ( X ) = (cid:114) I + I I I ∂∂ψ , ρ ¯ A ( X ) = 0 , ρ ¯ A ( X ) = 0 , ρ ¯ A ( X ) = 0 . The local expression of the Euler-Lagrange equations for the reduced Lagrangian system ¯ L : ¯ A → R is: ˙ ψ = (cid:114) I + I I I v , ˙ v = − (cid:114) I + I I I ∂V∂ψ , ˙ v = − (cid:115) I I ( I + I ) v v + 1 √ I + I v v , ˙ v = (cid:115) I I ( I + I ) v v − √ I + I v v , ˙ v = 0 . AMILTON-JACOBI EQUATION AND NONHOLONOMIC MECHANICS 25 From the two first equations we obtain the equation:¨ ψ = − I + I I I ∂V∂ψ . A section α : S → ¯ A ∗ , α ( ψ ) = ( ψ, α ( ψ ) , α ( ψ ) , α ( ψ ) , α ( ψ )), is a 1-cocycle, i.e. d ¯ A α = 0,if and only if α ( ψ ) = 0, α ( ψ ) = 0 and ∂α ∂ψ = 0. Therefore, the Hamilton-Jacobi equation d ¯ A ( E ¯ L ◦ ¯ G ◦ α ) = 0 is ∂V∂ψ + ∂α ∂ψ α = 0 . Thus, integrating we obtain 2 V ( ψ ) + ( α ( ψ )) = k with k constant. Therefore, α ( ψ ) = (cid:112) k − V ( ψ )and all the solutions of the Hamilton-Jacobi equation are of the form α ( ψ ) = ( ψ ; (cid:112) k − V ( ψ ) , , , k ) . with k constant.5.2. Mechanical systems subjected to linear nonholonomic constraints on a Lie algebroid. Let τ A : A → Q be a Lie algebroid over a manifold Q and denote by ([[ · , · ]] A , ρ A ) the Lie algebroidstructure on A .A mechanical system subjected to linear nonholonomic constraints on A is a pair( L, D ), where:(i) L : A → R is a Lagrangian function of mechanical type , that is, L ( a ) = 12 G ( a, a ) − V ( τ A ( a )) , for a ∈ A, and(ii) D is the total space of a vector subbundle τ D : D → Q of A . The vector subbundle D is saidto be the constraint subbundle .This kind of systems were considered in [7, 9, 14].We will denote by i D : D → A the canonical inclusion. We also consider the orthogonal decom-position A = D ⊕ D ⊥ and the associated orthogonal projectors P : A → D and Q : A → D ⊥ . Then,the solutions of the dynamical equations for the nonholonomic (constrained) system ( L, D ) are justthe integral curves of the vector field ξ ( L,D ) on D defined by ξ ( L,D ) = T P ◦ ξ L ◦ i D , where ξ L is the solution of the free dynamics (see Section 5.1) and T P : T A → T D is the tangentmap to the projector P .In fact, suppose that ( q i ) are local coordinates on an open subset U of Q and that { X B } = { X γ , X b } is a basis of sections of the vector bundle τ − A ( U ) → U such that { X γ } (respectively, { X b } ) is anorthonormal basis of sections of the vector subbundle τ − D ( U ) → U (respectively, τ − D ⊥ ( U ) → U ). We will denote by ( q i , v B ) = ( q i , v γ , v b ) the corresponding local coordinates on A . Then, the localequations defining the vector subbundle D are v b = 0 , for all b. Moreover, if ρ iB and C EBC are the local structure functions of A , we have that the local expression ofthe vector field ξ ( L,D ) is ξ ( L,D ) = ρ iγ v γ ∂∂q i − ( C νδγ v γ v ν + ρ iδ ∂V∂q i ) ∂∂v δ . (5.3)Thus, the dynamical equations for the constrained system ( L, D ) are˙ q i = ρ iγ v γ , ˙ v δ = − C νδγ v γ v ν − ρ iδ ∂V∂q i , v b = 0 . (5.4)On the other hand, the constrained connection ˇ ∇ : Γ( τ A ) × Γ( τ A ) → Γ( τ A ) associated with thesystem ( L, D ) is given by ˇ ∇ X Y = P ( ∇ G X Y ) + ∇ G X QY, for X, Y ∈ Γ( τ A ) . Therefore, if ˇΓ EBC are the coefficients of ˇ ∇ , we have thatˇΓ δγν = Γ δγν = 12 ( C νδγ + C γδν + C δγν ) , ˇΓ aγν = 0 . Consequently, Eqs. (5.4) are just the Lagrange-D’Alembert equations for the system ( L, D )considered in [7] (see also [9, 14]).Next, we will introduce a linear almost Poisson structure {· , ·} D ∗ on D ∗ .Denote by {· , ·} A ∗ the linear Poisson bracket on A ∗ induced by the Lie algebroid structure on A .Then, { ϕ, ψ } D ∗ = { ϕ ◦ i ∗ D , ψ ◦ i ∗ D } A ∗ ◦ P ∗ , (5.5)for ϕ, ψ ∈ C ∞ ( D ∗ ), where i ∗ D : A ∗ → D ∗ and P ∗ : D ∗ → A ∗ are the dual maps of the monomorphism i D : D → A and the projector P : D → A , respectively.It is easy to prove that {· , ·} D ∗ is a linear almost Poisson bracket on D ∗ . Moreover, if ( q i , p B ) =( q i , p γ , p b ) are the dual coordinates of ( q i , v B ) = ( q i , v γ , v b ) on A ∗ then it is clear that ( q i , p γ ) are localcoordinates on D ∗ and, in addition, the local expressions of i ∗ D and P ∗ are i ∗ D ( q i , p γ , p b ) = ( q i , p γ ) , P ∗ ( q i , p γ ) = ( q i , p γ , . (5.6)Thus, from (2.1), (5.5) and (5.6), we have that { ϕ, ψ } D ∗ = ρ iγ ( ∂ϕ∂q i ∂ψ∂p γ − ∂ϕ∂p γ ∂ψ∂q i ) − C γβδ p γ ∂ϕ∂p β ∂ψ∂p δ , (5.7)for ϕ, ψ ∈ C ∞ ( D ∗ ).On the other hand, one may introduce a linear Poisson bracket {· , ·} A on A in such a way thatthe vector bundle map (cid:91) G : A → A ∗ is a Poisson isomorphism, when on A ∗ we consider the linearPoisson structure {· , ·} A ∗ . Since the local expression of (cid:91) G is (cid:91) G ( q i , v B ) = ( q i , v B )we deduce that the local expression of the linear Poisson bracket {· , ·} A is { ¯ ϕ, ¯ ψ } A = ρ iB ( ∂ ¯ ϕ∂q i ∂ ¯ ψ∂v B − ∂ ¯ ϕ∂v B ∂ ¯ ψ∂q i ) − C EBC v E ∂ ¯ ϕ∂v B ∂ ¯ ψ∂v C , AMILTON-JACOBI EQUATION AND NONHOLONOMIC MECHANICS 27 for ¯ ϕ, ¯ ψ ∈ C ∞ ( A ).Using the bracket {· , ·} A , one may define a linear almost Poisson bracket {· , ·} nh on D as follows.If ˜ ϕ and ˜ ψ are real C ∞ -functions on D then { ˜ ϕ, ˜ ψ } nh = { ˜ ϕ ◦ P, ˜ ψ ◦ P } A ◦ i D . We have that { ˜ ϕ, ˜ ψ } nh = ρ iγ ( ∂ ˜ ϕ∂q i ∂ ˜ ψ∂v γ − ∂ ˜ ϕ∂v γ ∂ ˜ ψ∂q i ) − C γβδ v γ ∂ ˜ ϕ∂v β ∂ ˜ ψ∂v δ . (5.8)Thus, a direct computation proves that {· , ·} nh is just the nonholonomic bracket introduced in[7]. Note that, using (5.3) and (5.8), we obtain that ξ ( L,D ) is the Hamiltonian vector field of thefunction ( E L ) | D with respect to the nonholonomic bracket {· , ·} nh , i.e.,˙˜ ϕ = ξ ( L,D ) ( ˜ ϕ ) = { ˜ ϕ, ( E L ) | D } nh , for ˜ ϕ ∈ C ∞ ( D ) (see also [7]).Moreover, if G D is the restriction of the bundle metric G to D and (cid:91) G D : D → D ∗ is the correspondingvector bundle isomorphism then, from (5.7) and (5.8), we deduce that { ϕ ◦ (cid:91) G D , ψ ◦ (cid:91) G D } nh = { ϕ, ψ } D ∗ ◦ (cid:91) G D , for ϕ, ψ ∈ C ∞ ( D ∗ ) . For this reason, {· , ·} D ∗ will also be called the nonholonomic bracket associated with theconstrained system ( L, D ).We will denote by ([[ · , · ]] D , ρ D ) (respectively, d D ) the corresponding skew-symmetric algebroid struc-ture (respectively, almost differential) on the vector bundle τ D : D → Q and by G D : D ∗ → D theinverse morphism of (cid:91) G D : D → D ∗ .Then, from (2.2) and (5.5), it follows that[[ X, Y ]] D = P [[ i D ◦ X, i D ◦ Y ]] A , ρ D ( X ) = ρ A ( i D ◦ X ) , (5.9)for X, Y ∈ Γ( τ D ). Therefore, using (2.5), we have that d D α = Λ k i ∗ D ( d A ( P ∗ ◦ α )) , for α ∈ Γ(Λ k τ D ∗ ) . (5.10)On the other hand, if α : Q → D ∗ is a section of the vector bundle τ D ∗ : D ∗ → Q one may considerthe vector field ξ ( L,D ) ,α on Q given by ξ ( L,D ) α ( q ) = ( T G D ( α ( q )) τ D )( ξ ( L,D ) ( G D ( α ( q )))) , for q ∈ Q. (5.11) Corollary 5.6. Let α : Q → D ∗ be a -cocycle of the skew-symmetric algebroid ( D, [[ · , · ]] D , ρ D ) , thatis, d D α = 0 . Then, the following conditions are equivalent: (i) If c : I → Q is an integral curve of the vector field ξ ( L,D ) α on Q we have that G D ◦ α ◦ c : I → D is a solution of the Lagrange-D’Alembert equations for the constrained system ( L, D ) . (ii) α satisfies the nonholonomic Hamilton-Jacobi equation d D (( E L ) | D ◦ G D ◦ α ) = 0 . If, additionally, H ( d D ) (cid:39) R (or the skew-symmetric algebroid ( D, [[ · , · ]] D , ρ D ) is completelynonholonomic and Q is connected) then conditions (i) and (ii) are equivalent to (iii) ( E L ) | D ◦ G D ◦ α = constant . Proof. Denote by h ( L,D ) : D ∗ → R the Hamiltonian function on D ∗ given by h ( L,D ) = ( E L ) | D ◦ G D ,by Λ D ∗ the linear almost Poisson 2-vector on D ∗ and by H Λ D ∗ h ( L,D ) the Hamiltonian vector field of h ( L,D ) with respect to Λ D ∗ . Then, the vector fields ξ ( L,D ) and H Λ D ∗ h ( L,D ) on D and D ∗ , respectively, are (cid:91) G D -related. Thus, from (5.11) and since τ D ∗ ◦ (cid:91) G D = τ D , it follows that H Λ D ∗ h ( L,D ) α ( q ) = ( T α ( q ) τ D ∗ )( H Λ D ∗ h ( L,D ) ( α ( q ))) = ξ ( L,D ) α ( q ) , that is, the vector fields H Λ D ∗ h ( L,D ) α and ξ ( L,D ) α are equal.Moreover, if σ : I → D is a curve on D , we have that σ is a solution of the Lagrange-D’Alembertequations for the constrained system ( L, D ) if and only if (cid:91) G D ◦ σ : I → D ∗ is a solution of theHamilton equations for h ( L,D ) .Therefore, using Theorem 4.1, we deduce that conditions (i) and (ii) are equivalent.In addition, if H ( d D ) (cid:39) R (or if ( D, [[ · , · ]] D , ρ D ) is completely nonholonomic and Q is connected)then, from Corollary 4.6, it follows that conditions (i), (ii) and (iii) are equivalent. (cid:3) Remark 5.7. Let D be the annihilator of D and I ( D ) be the algebraic ideal generated by D .Thus, a section ν of the vector bundle Λ k A ∗ → Q belongs to I ( D ) if ν ( q )( v , . . . , v k ) = 0 , for all q ∈ Q and v , . . . , v k ∈ D q . Now, let Z ( τ D ∗ ) be the set defined by Z ( τ D ∗ ) = { α ∈ Γ( τ D ∗ ) /d D α = 0 } and ˜ Z ( τ ( D ⊥ ) ) be the set given by˜ Z ( τ ( D ⊥ ) ) = { ˜ α ∈ Γ( τ ( D ⊥ ) ) /d A ˜ α ∈ I ( D ) } where ( D ⊥ ) is the annihilator of the orthogonal complement D ⊥ of D and τ ( D ⊥ ) : ( D ⊥ ) → Q isthe corresponding vector bundle projection. Then, using (5.10), we deduce that the map Z ( τ D ∗ ) → ˜ Z ( τ ( D ⊥ ) ) , α → P ∗ ◦ α defines a bijection from Z ( τ D ∗ ) on ˜ Z ( τ ( D ⊥ ) ). In fact, the inverse map is given by˜ Z ( τ ( D ⊥ ) ) → Z ( τ D ∗ ) , ˜ α → i ∗ D ◦ ˜ α. On the other hand, if f is a real C ∞ -function on Q then d D f = 0 ⇐⇒ ( d A f )( Q ) ⊆ D . (cid:5) Let ( L, D ) (respectively, ( ¯ L, ¯ D )) be a nonholonomic system on a Lie algebroid τ A : A → Q (respectively, τ ¯ A : ¯ A → ¯ Q ) and ( ˜ F , F ) be a linear almost Poisson morphism between the almostPoisson manifolds ( D ∗ , {· , ·} D ∗ ) and ( ¯ D ∗ , {· , ·} ¯ D ∗ ) such that:(i) F : Q → ¯ Q is a surjective map.(ii) For each q ∈ Q , the linear map ˜ F q = ˜ F | D ∗ q : D ∗ q → ¯ D ∗ F ( q ) satisfies the following conditions G ¯ D ∗ ( ˜ F q ( β ) , ˜ F q ( β (cid:48) )) = G D ∗ ( β, β (cid:48) ) , for β, β (cid:48) ∈ D ∗ q ,F ( q ) = F ( q (cid:48) ) = ⇒ ˜ F q ( D ∗ q ) = ˜ F q (cid:48) ( D ∗ q (cid:48) ) , where G D ∗ (respectively, G ¯ D ∗ ) is the bundle metric on D ∗ (respectively, ¯ D ∗ ). AMILTON-JACOBI EQUATION AND NONHOLONOMIC MECHANICS 29 (iii) If V : Q → R (respectively, ¯ V : ¯ Q → R ) is the potential energy for the nonholonomic systemon A (respectively, ¯ A ) we have that ¯ V ◦ F = V .Then, we deduce that ( ˜ F , F ) is a Hamiltonian morphism between the Hamiltonian systems ( D, {· , ·} D ∗ ,h ( L,D ) ) and ( ¯ D, {· , ·} ¯ D ∗ , h (¯ L, ¯ D ) ), where h ( L,D ) (respectively, h (¯ L, ¯ D ) ) is the constrained Hamiltonianfunction on D ∗ (respectively, ¯ D ∗ ) associated with the nonholonomic system ( L, D ) (respectively,( ¯ L, ¯ D )).Moreover, using Theorem 4.12, we conclude that Corollary 5.8. (i) If α : Q → D ∗ is a -cocycle for the skew-symmetric algebroid D ( d D α = 0 ),it satisfies the Hamilton-Jacobi equation d D (( E L ) | D ◦ G D ◦ α ) = 0 (5.12) (respectively, the strongest condition ( E L ) | D ◦ G D ◦ α = constant ) and it is ( ˜ F , F ) -relatedwith ¯ α ∈ Γ( τ ¯ D ∗ ) then d ¯ D ¯ α = 0 and ¯ α is a solution of the Hamilton-Jacobi equation d ¯ D (( E ¯ L ) | ¯ D ◦ G ¯ D ◦ ¯ α ) = 0 (5.13) (respectively, ¯ α satisfies the strongest condition ( E ¯ L ) | ¯ D ◦ G ¯ D ◦ ¯ α = constant ). (ii) If ¯ α : ¯ Q → ˜ F ( D ∗ ) ⊆ ¯ D ∗ is a -cocycle for the skew-symmetric algebroid ¯ D ( d ¯ D ¯ α = 0 ) andit satisfies the Hamilton-Jacobi equation (5.13) (respectively, the strongest condition ( E ¯ L ) | ¯ D ◦ G ¯ D ◦ ¯ α = constant ) then d D α = 0 and α is a solution of the Hamilton Jacobi equation(5.12) (respectively, α satisfies the strongest condition ( E L ) | D ◦ G D ◦ α = constant ). Here, α : Q → D ∗ is the section of τ D ∗ : D ∗ → Q characterized by the condition ˜ F ◦ α = ¯ α ◦ F . The particular case A = T Q . Let L : T Q → R be a Lagrangian function of mechanical typeon the standard Lie algebroid τ T Q : T Q → Q , that is, L ( v ) = 12 g ( v, v ) − V ( τ Q ( v )) , for v ∈ T Q, where g is a Riemannian metric on Q and V : Q → R is a real C ∞ -function on Q . Suppose alsothat D is a distribution on Q . Then, the pair ( L, D ) is a mechanical system subjected to linearnonholonomic constraints on the standard Lie algebroid τ T Q : T Q → Q .Note that, in this case, the linear Poisson structure on A ∗ = T ∗ Q is induced by the canonicalsymplectic structure on T ∗ Q . Moreover, the corresponding nonholonomic bracket {· , ·} D ∗ on D ∗ was considered by several authors or, alternatively, other almost Poisson structures (on D or on (cid:91) g ( D ) ⊆ A ∗ = T ∗ Q ) which are isomorphic to {· , ·} D ∗ also were obtained by several authors (see[5, 19, 22, 39]).Now, denote by g : T ∗ Q → T Q (respectively, g D : D ∗ → D ) the inverse morphism of themusical isomorphism (cid:91) g : T Q → T ∗ Q (respectively, (cid:91) g D : D → D ∗ ) induced by the Riemannianmetric g (respectively, by the restriction g D of g to D ), by d the standard exterior differential on Q (that is, d = d T Q is the differential of the Lie algebroid τ T Q : T Q → Q ), by ξ ( L,D ) ∈ X ( D ) thesolution of the nonholonomic dynamics and by ξ ( L,D ) α ∈ X ( Q ) its projection on Q , α being a sectionof the vector bundle τ D ∗ : D ∗ → Q (see (5.11)). Using this notation, Corollary 5.6 and Remark 5.7,we deduce the following result Corollary 5.9. Let α : Q → D ∗ be a section of the vector bundle τ D ∗ : D ∗ → Q such that d ( P ∗ ◦ α ) ∈ I ( D ) . Then, the following conditions are equivalent: (i) If c : I → Q is an integral curve of the vector field ξ ( L,D ) α on Q we have that g D ◦ α ◦ c : I → D is a solution of the Lagrange-D’Alembert equations for the constrained system ( L, D ) . (ii) d (( E L ) | D ◦ g D ◦ α )( Q ) ⊆ D . Remark 5.10. As we know, the Legendre transformation associated with the Lagrangian function L : T Q → R is the musical isomorphism (cid:91) g : T Q → T ∗ Q . Moreover, it is clear that X ( Q ) ⊆ D ,where X is the vector field on Q given by X = g D ◦ α . Thus, Corollary 5.9 is a consequence ofsome results which were proved in [20] (see Theorem 4.3 in [20]). On the other hand, if H ( d D ) (cid:39) R (or if Q is connected and the distribution D is completely nonholonomic in the sense of Vershik andGershkovich [44]) then (i) and (ii) in Corollary 5.9 are equivalent to the condition( E L ) | D ◦ g D ◦ α = constant . A Hamiltonian version of this last result was proved by Ohsawa and Bloch [31] (see Theorem 3.1 in[31]). (cid:5) Remark 5.11. Previous approaches. There exists some different attempts in the literature ofextending the classical Hamilton-Jacobi equation for the case of nonholonomic constraints [10, 33, 36,40, 41, 42, 43]). These attempts were non-effective or very restrictive (and even erroneous), because,in many of them, they try to adapt the standard proof of the Hamilton-Jacobi equations for systemswithout constraints, using Hamilton’s principle. See [37] for a detailed discussion on the topic.To fix ideas, consider a lagrangian system L : T Q −→ R of mechanical type, that is, L ( v q ) = G ( v q , v q ) − V ( q ), for v q ∈ T q Q , and nonholonomic constraints determined by a distribution D of Q ,whose annihilator is D = span { µ bi dq i } .The idea of many of these previous approaches consist in looking for a function S : Q −→ R calledthe characteristic function which permits characterize the solutions of the nonholonomic problem.For it, define first the generalized momenta p i = ∂S∂q i + λ b µ bi , which satisfy the constraint equations G ij p i µ aj = 0. These last conditions univocally determine λ b asfunctions of q and ∂S/∂q and therefore we find the momenta as functions p i = p i ( q i , ∂S∂q i ) . (5.14)By inserting these expressions for the generalized momenta in the Hamiltonian of the system, weobtain a version of the Hamilton-Jacobi equation (in its time-independent version): H ( q i , p i ) = ˜ H ( q i , ∂S∂q i ) = constant. (5.15)However, if we start with a curve c : I → Q satisfying the differential equations˙ c i ( t ) = ∂H∂p i ( c j ( t ) , ∂S∂q j ( c ( t )) + λ b µ bj ) (5.16)in general, it is not true that the curve γ ( t ) = ( c i ( t ) , p i ( t )) is a solution of the nonholonomic equations.This is trivially checked since from Equation (5.15) we deduce that:0 = ∂H∂q i + ∂H∂p j (cid:34) ∂ S∂q i ∂q j + ∂λ b ∂q i µ bj + λ b ∂µ bj ∂q i (cid:35) (5.17) AMILTON-JACOBI EQUATION AND NONHOLONOMIC MECHANICS 31 but, on the other hand, ˙ p i = ddt (cid:20) ∂S∂q i + λ b µ bi (cid:21) , = ˙ q j (cid:20) ∂ S∂q i ∂q j + ∂λ b ∂q j µ bi + λ b ∂µ bi ∂q j (cid:21) . (5.18)Substituting Equation (5.17) in Equation (5.18), a curve γ ( t ) = ( c i ( t ) , p i ( t )) satisfying (5.16) issolution of the nonholonomic equations (that is, ˙ p i = − ∂H∂q i +Λ b µ bi ) if it verifies the following condition: λ b (cid:32) ∂µ bj ∂q i ˙ q j − ∂µ bi ∂q j ˙ q j (cid:33) δq i = 0 , δq ∈ D q . (5.19)It is well-known (see [36, 37]) that condition (5.19) takes place when the solutions of the nonholo-nomic problem are also of variational type. However, nonholonomic dynamics is not, in general, ofvariational kind (see [8, 24, 26]). Indeed, a relevant difference with the unconstrained mechanicalsystems is that a nonholonomic system is not Hamiltonian in the standard sense since the dynamicsis obtained from an almost Poisson bracket, that is, a bracket not satisfying the Jacobi identity (see[5, 19, 22, 39]). (cid:5) An explicit example: The two-wheeled carriage (see [30]). The system has configuration space Q = SE (2) × T , where SE (2) represents the rigid motions in the plane and T the angles of rotationof the left and right wheels. We use standard coordinates ( x, y, θ, ψ , ψ ) ∈ SE (2) × T . Imposing theconstraints of no lateral sliding and no sliding on both wheels, one gets the following nonholonomicconstraints: ˙ x sin θ − ˙ y cos θ = 0 , ˙ x cos θ + ˙ y sin θ + r ˙ θ + a ˙ ψ = 0 , ˙ x cos θ + ˙ y sin θ − r ˙ θ + a ˙ ψ = 0 , where a is the radius of the wheels and r is the half the length of the axle.Assuming, for simplicity, that the center of mass of the carriage is situated on the center of theaxle the Lagrangian is given by: L = 12 m ˙ x + 12 m ˙ y + 12 J ˙ θ + 12 C ˙ ψ + 12 C ˙ ψ , where m is the mass of the system, J the moment of inertia when it rotates as a whole about thevertical axis passing through the point ( x, y ) and C the axial moment of inertia. Note that L is thekinetic energy associated with the Riemannian metric g on Q given by g = m ( dx + dy ) + J dθ + Cdψ + Cdψ . The constraints induce the distribution D locally spanned by the following g -orthonormal vectorfields X = 1Λ (cid:18) r ∂∂ψ − a ∂∂θ − ar cos θ ∂∂x − ar sin θ ∂∂y (cid:19) ,X = 1Λ (cid:18) a ( J − m r ) ∂∂ψ + ( a J + 4 Cr + a m r ) ∂∂ψ + ar (2 C + m a ) ∂∂θ − a ( a J + 2 Cr ) cos θ ∂∂x − a ( a J + 2 Cr ) sin θ ∂∂y (cid:19) . where Λ = (cid:112) Cr + a J + am r Λ = (cid:112) ( a J + 2 Cr )(2 C + m a )( a J + 4 Cr + a r m )We will denote by ( x, y, θ, ψ , ψ , v , v ) the local coordinates on D induced by the basis { X , X } .In these coordinates, the restriction, L | D : D (cid:55)−→ R , of L to D is: L | D = 12 (( v ) + ( v ) ) . The distribution D ⊥ orthogonal to D is generated by D ⊥ = { X = tan θ ∂∂x − ∂∂y , X = J Sec( θ ) rm ∂∂x + ∂∂θ + aJCr ∂∂ψ , X = 2 C Sec( θ ) am ∂∂x + ∂∂ψ + ∂∂ψ } Moreover, since the standard Lie bracket [ X , X ] of the vector fields X and X is orthogonal to D , it follows that (see (5.9))[[ X , X ]] D = 0 , ρ D ( X ) = X , ρ D ( X ) = X , (5.20)where ([[ · , · ]] D , ρ D ) is the skew-symmetric algebroid structure on the vector bundle τ D : D → Q .The local expression of the vector field ξ ( L,D ) is: ξ ( L,D ) = (cid:18) rv Λ + a ( J − m r ) v Λ (cid:19) ∂∂ψ + ( a J + 4 Cr + a m r ) v Λ ∂∂ψ + (cid:18) ar (2 C + m a ) v Λ − av Λ (cid:19) ∂∂θ − (cid:18) arv cos θ Λ + a ( a J + 2 Cr ) v cos θ Λ (cid:19) ∂∂x − (cid:18) arv sin θ Λ + a ( a J + 2 Cr ) v sin θ Λ (cid:19) ∂∂y Furthermore, if { X , X } is the dual basis of { X , X } and α : Q → D ∗ is a section of the vectorbundle τ D ∗ : D ∗ → Q α = α X + α X , with α , α ∈ C ∞ ( Q )then α is a 1-cocycle ⇐⇒ X ( α ) − X ( α ) = 0 . In particular, taking α = K X + K X , with K , K ∈ R trivially is satisfied the 1-cocycle condition. AMILTON-JACOBI EQUATION AND NONHOLONOMIC MECHANICS 33 In addition, since E L = L , we deduce that( E L ) | D = 12 (( v ) + ( v ) )which implies that ( E L ) | D ◦ g D ◦ α = K + K = constant . Thus, using Corollary 5.6, we conclude that to integrate the nonholonomic mechanical system ( L, D )is equivalent to find the integral curves of the vector field on Q = S × S × R given by ξ ( L,D ) ,α = K X + K X . which are easily obtained.It is also interesting to observe that, in this particular example,Lie ∞ ( D ) = { adψ − adψ + 2 rdθ } and, thus, D is not completely nonholonomic. From Theorem 3.4 it is necessary to restrict the initialnonholonomic system to the orbits of D , that in this case are L k = { ( x, y, θ, ψ , ψ ) ∈ SE (2) × T | a ( ψ − ψ ) + 2 rθ = k, with k ∈ R } to obtain a completely nonholonomic skew-symmetric algebroid structure on the vector bundle τ D Lk : D L k → L k . Note that on L k we can use, for instance, coordinates ( x, y, ψ , ψ ).5.2.2. The particular case ¯ A = T Q/G . Let F : Q → ¯ Q = Q/G be a principal G -bundle and τ ¯ A = τ T Q/G : ¯ A = T Q/G → ¯ Q = Q/G be the Atiyah algebroid associated with the principal bundle(see Section 5.1).Suppose that g is a G -invariant Riemannian metric on Q , that V : Q → R is a G -invariant real C ∞ -function and that D is a G -invariant distribution on Q . Then, we may consider the correspondingnonholonomic mechanical system ( L, D ) on the standard Lie algebroid τ A = τ T Q : A = T Q → Q .Denote by ξ ( L,D ) ∈ X ( D ) the nonholonomic dynamics for the system ( L, D ) and by {· , ·} D ∗ thenonholonomic bracket on D ∗ .The Riemannian metric g and the function V : Q → R induce a bundle metric ¯ G on the Atiyahalgebroid τ ¯ A = τ T Q/G : ¯ A = T Q/G → ¯ Q = Q/G and a real C ∞ -function ¯ V : ¯ Q → R on ¯ Q such that¯ V ◦ F = V , where F : Q → ¯ Q = Q/G is the canonical projection. Moreover, the space of orbits ¯ D of the action of G on D is a vector subbundle of the Atiyah algebroid τ ¯ A = τ T Q/G : ¯ A = T Q/G → ¯ Q = Q/G . Thus, we may consider the corresponding nonholonomic mechanical system ( ¯ L, ¯ D ) on¯ A = T Q/G .Let ¯ F : A = T Q → ¯ A = T Q/G be the canonical projection. Then, ( ¯ F , F ) is a fiberwise bijectivemorphism of Lie algebroids and ¯ F ( D ) = ¯ D . Therefore, using some results in [7] (see Theorem4.6 in [7]) we deduce that the vector field ξ ( L,D ) is ¯ F D -projectable on the nonholonomic dynamics ξ (¯ L, ¯ D ) ∈ X ( ¯ D ) of the system ( ¯ L, ¯ D ). Here, ¯ F D : D → ¯ D = D/G is the canonical projection.On the other hand, if P : A = T Q → D and ¯ P : ¯ A = T Q/G → ¯ D = D/G are the orthogonalprojectors then it is clear that ¯ F D ◦ P = ¯ P ◦ ¯ F which implies that ˜ F ◦ P ∗ = ˜ F D ◦ ¯ P ∗ , (5.21) where ˜ F : A ∗ = T ∗ Q → ¯ A ∗ (cid:39) T ∗ Q/G and ˜ F D : D ∗ → ¯ D ∗ (cid:39) D ∗ /G are the canonical projections.Moreover, if on ¯ A ∗ we consider the linear Poisson structure induced by the Atiyah algebroid τ ¯ A = τ T Q/G : ¯ A = T Q/G → ¯ Q = Q/G then, as we know, ˜ F : A ∗ = T ∗ Q → ¯ A ∗ (cid:39) T ∗ Q/G is a Poissonmorphism. Thus, using this fact, (5.5) and (5.21), we deduce the following result Proposition 5.12. The pair ( ˜ F D , F ) is a linear almost Poisson morphism, when on D ∗ and ¯ D ∗ weconsider the almost Poisson structures induced by the nonholonomic brackets {· , ·} D ∗ and {· , ·} ¯ D ∗ ,respectively. Note that Proposition 5.12 characterizes the nonholonomic bracket {· , ·} ¯ D ∗ .We also note that the linear map ( ˜ F D ) q = ( ˜ F D ) | D ∗ q : D ∗ q → ¯ D ∗ F ( q ) (cid:39) ( D ∗ /G ) F ( q ) is a linear isometry,for all q ∈ Q . Therefore, from Remark 5.7 and Corollary 5.8, it follows Corollary 5.13. Let S the set of the -cocycles ¯ α of the skew-symmetric algebroid τ ¯ D = τ D/G : ¯ D = D/G → ¯ Q = Q/G which are solution of the nonholonomic Hamilton-Jacobi equation d ¯ D (( E ¯ L ) | ¯ D ◦ G ¯ D ◦ ¯ α ) = 0 (respectively, which satisfy the strongest condition ( E ¯ L ) | ¯ D ◦ G ¯ D ◦ ¯ α = constant ). Then, there existsa one-to-one correspondence between S and the following sets: (i) The set of the G -invariant -cocycles α of the skew-symmetric algebroid τ D : D → Q whichare solutions of the nonholonomic Hamilton-Jacobi equation d D (( E L ) | D ◦ G D ◦ α ) = 0 (respectively, which satisfy the strongest condition ( E L ) | D ◦ G D ◦ α = constant ). (ii) The set of the G -invariant -forms γ : Q → ( D ⊥ ) ⊆ T ∗ Q on Q which satisfy the followingconditions dγ ∈ I ( D ) and d ( E L ◦ g ◦ γ )( Q ) ⊆ D (respectively, which satisfy the strongest conditions dγ ∈ I ( D ) and E L ◦ g ◦ γ = constant ).An explicit example: The snakeboard .The snakeboard is a modified version of the traditional skateboard, where the rider uses his ownmomentum, coupled with the constraints, to generate forward motion. The configuration manifoldis Q = SE (2) × T with coordinates ( x, y, θ, ψ, φ ) (see [4, 21]). φ − θr ψ − θ θ φ − θ The system is described by a Lagrangian L ( q, ˙ q ) = 12 m ( ˙ x + ˙ y ) + 12 ( J + 2 J ) ˙ θ + 12 J ( ˙ θ + ˙ ψ ) + J ˙ φ AMILTON-JACOBI EQUATION AND NONHOLONOMIC MECHANICS 35 where m is the total mass of the board, J > J > J > r . For simplicity, as in [21], we assume that J + J + 2 J = mr .The inertia matrix representing the kinetic energy of the metric g on Q defined by the snakeboardis g = mdx + mdy + mr dθ + J dθ ⊗ dψ + J dψ ⊗ dθ + J dψ + 2 J dφ . Since the wheels are not allowed to slide in the sideways direction, we impose the constraints − ˙ x sin( θ + φ ) + ˙ y cos( θ + φ ) − r ˙ θ cos φ = 0 − ˙ x sin( θ − φ ) + ˙ y cos( θ − φ ) + r ˙ θ cos φ = 0 . To avoid singularities of the distribution defined by the previous constraints we will assume, in thesequel, that φ (cid:54) = ± π/ a = − r (cos φ cos( θ − φ ) + cos φ cos( θ + φ )) = − r cos φ cos θb = − r (cos φ sin( θ − φ ) + cos φ sin( θ + φ )) = − r cos φ sin θc = sin(2 φ ) . The constraint subbundle τ D : D (cid:55)−→ Q is D = span (cid:26) ∂∂ψ , ∂∂φ , a ∂∂x + b ∂∂y + c ∂∂θ (cid:27) . The Lagrangian function and the constraint subbundle are left-invariant under the SE (2) action:Φ g ( q ) = ( α + x cos γ − y sin γ, β + x sin γ + y cos γ, γ + θ, ψ, φ )where g = ( α, β, γ ) ∈ SE (2).We have a principal bundle structure F : Q −→ ¯ Q where ¯ Q = ( SE (2) × T ) /SE (2) (cid:39) T , beingits vertical bundle V F = span (cid:26) ∂∂x , ∂∂y , ∂∂θ (cid:27) . We have that S = D ∩ V F = span (cid:26) Y = a ∂∂x + b ∂∂y + c ∂∂θ (cid:27) and therefore, S ⊥ ∩ D = span (cid:26) Y = ∂∂φ , Y = ∂∂ψ − J ck Y (cid:27) = span (cid:26) Y = ∂∂φ , Y = ∂∂ψ − J mr (tan φ ) Y (cid:27) where k = m ( a + b + c r ) = 4 mr (cos φ ) (away form φ = ± π/ { ξ , ξ , ξ } is the canonical basis of se (2) then Y = ∂∂ψ − J sin φmr (cid:104) − r (cos φ ) ←− ξ + (sin φ ) ←− ξ (cid:105) Y = − r (cos φ ) ←− ξ + (sin 2 φ ) ←− ξ where ←− ξ i ( i = 1 , , 3) is the left-invariant vector field of SE (2) such that ←− ξ i ( e ) = ξ i , e being theidentity element of SE (2).Next, we will denote by { X , X , X } the g -orthonormal basis of D given by X = 1 √ J ∂∂φ ,X = 1 (cid:112) f ( φ ) (cid:18) ∂∂ψ − J sin φmr (cid:104) − r (cos φ ) ←− ξ + (sin φ ) ←− ξ (cid:105)(cid:19) X = 1 √ m (cid:20) − (cos φ ) ←− ξ + 1 r (sin φ ) ←− ξ (cid:21) , where f ( φ ) = J − J sin φmr . The vector fields { X , X } describe changes in the internal angles φ and ψ , while X represents the instantaneous rotation when the internal angles are fixed.Consider now the corresponding Atiyah algebroid T Q/SE (2) (cid:39) ( T T × T SE (2)) /SE (2) −→ ¯ Q = T . Using the left translations on SE (2), we have that the tangent bundle of SE (2) may be identifiedwith the product manifold SE (2) × se (2) and therefore the Atiyah algebroid is identified with thevector bundle ˜ τ T = τ ¯ A : ¯ A = T T × se (2) −→ T . The canonical basis of τ ¯ A : T T × se (2) −→ T is (cid:26) ∂∂ψ , ∂∂φ , ξ , ξ , ξ (cid:27) . The anchor map and the linear bracket of the Lie algebroid τ ¯ A : T T × se (2) −→ T is given by ρ ¯ A ( ∂∂ψ ) = ∂∂ψ , ρ ¯ A ( ∂∂φ ) = ∂∂φ , ρ ¯ A ( ξ i ) = 0 , i = 1 , , ξ , ξ ]] ¯ A = − ξ , [[ ξ , ξ ]] ¯ A = ξ , being equal to zero the rest of the fundamental Lie brackets.We select the orthonormal basis of sections, { X (cid:48) , X (cid:48) , X (cid:48) , X (cid:48) , X (cid:48) } , where X (cid:48) = 1 √ J ∂∂φ ,X (cid:48) = 1 (cid:112) f ( φ ) (cid:18) ∂∂ψ − J sin φmr [ − r (cos φ ) ξ + (sin φ ) ξ ] (cid:19) X (cid:48) = 1 √ m (cid:20) − (cos φ ) ξ + 1 r (sin φ ) ξ (cid:21) , and { X (cid:48) , X (cid:48) } is an orthonormal basis of sections of the orthogonal complement to ¯ D , ¯ D ⊥ , withrespect to the induced bundle metric G ¯ A .Taking the induced coordinates ( ψ, φ, v , v , v , v , v ) on T T × se (2) by this basis of sections, wededuce that the space of orbits ¯ D of the action of SE (2) on D has as local equations, v = 0 and v = 0, being a basis of sections of ¯ D , { X (cid:48) , X (cid:48) , X (cid:48) } . Moreover, in these coordinates the reduced AMILTON-JACOBI EQUATION AND NONHOLONOMIC MECHANICS 37 Lagrangian ¯ L : T T × se (2) −→ R is¯ L = 12 (cid:0) ( v ) + ( v ) + ( v ) + ( v ) + ( v ) (cid:1) . Now, we consider the reduced nonholonomic mechanical system ( ¯ L, ¯ D ).After, some straightforward computations we deduce that[[ X (cid:48) , X (cid:48) ]] ¯ D = − J cos φr (cid:112) J mf ( φ ) X (cid:48) , [[ X (cid:48) , X (cid:48) ]] ¯ D = J cos φr (cid:112) J mf ( φ ) X (cid:48) , [[ X (cid:48) , X (cid:48) ]] ¯ D = 0 . Therefore, the non-vanishing structure functions are: C = − C = − J cos φr (cid:112) J mf ( φ ) , C = − C = J cos φr (cid:112) J mf ( φ ) . Moreover, ρ ¯ D ( X (cid:48) ) = 1 √ J ∂∂φ , ρ ¯ D ( X (cid:48) ) = 1 (cid:112) f ( φ ) ∂∂ψ , ρ ¯ D ( X (cid:48) ) = 0 . This shows that ρ ¯ D ( ¯ D ) = T q T and then the skew-symmetric algebroid ¯ D −→ T is completelynonholonomic.The local expression of the vector field ξ (¯ L, ¯ D ) is ξ (¯ L, ¯ D ) = v √ J ∂∂φ + v (cid:112) f ( φ ) ∂∂ψ − J cos φr (cid:112) J mf ( φ ) v v ∂∂v + J cos φr (cid:112) J mf ( φ ) v v ∂∂v Let { ( X (cid:48) ) , ( X (cid:48) ) , ( X (cid:48) ) } be the dual basis of ¯ D ∗ . It induces a local coordinate system: ( φ, ψ, p , p , p )on ¯ D ∗ and, therefore, the non-vanishing terms of the nonholonomic bracket are: { φ, p } ¯ D ∗ = 1 √ J , { ψ, p } ¯ D ∗ = 1 (cid:112) f ( φ ) , { p , p } ¯ D ∗ = J cos φr (cid:112) J mf ( φ ) p , { p , p } ¯ D ∗ = − J cos φr (cid:112) J mf ( φ ) p . Now, we study the Hamilton-Jacobi equations for the snakeboard system. A section α : T −→ ¯ D ∗ , α = α ( φ, ψ )( X (cid:48) ) + α ( φ, ψ )( X (cid:48) ) + α ( φ, ψ )( X (cid:48) ) , is a 1-cocycle ( d ¯ D α = 0) if and only if:0 = 1 √ J ∂α ∂φ − (cid:112) f ( φ ) ∂α ∂ψ + J cos φr (cid:112) J mf ( φ ) α (5.22)0 = 1 √ J ∂α ∂φ − J cos φr (cid:112) J mf ( φ ) α (5.23)0 = 1 (cid:112) f ( φ ) ∂α ∂ψ . (5.24) Finally, since the skew-symmetric algebroid is completely nonholonomic, the Hamilton-Jacobiequation is rewritten as ( α ( φ, ψ )) + ( α ( φ, ψ )) + ( α ( φ, ψ )) = constant (5.25)Now, we will use this equation for studying explicit solutions for the snakeboard, showing theavailability of our methods for obtaining new insights in nonholonomic dynamics.From Equation (5.24) we obtain that α = α ( φ ). Then it is clear that also α = α ( φ ). Assumethat α = constant. Therefore, Equations (5.22) and (5.23) are now, in this case, a system of ordinarydifferential equations: 0 = dα dφ + J cos φr (cid:112) mf ( φ ) α (5.26)0 = dα dφ − J cos φr (cid:112) mf ( φ ) α . (5.27)Moreover, observe that all the solutions of these equations automatically satisfy Equation (5.25)since α dα dφ + α dα dφ = 0 and α = constantSolving explicitly the system of equations (5.26) and (5.27) we obtain that α ( φ ) = C (cid:112) f ( φ ) + J C r √ m sin φα ( φ ) = J C r √ m sin φ − C (cid:112) f ( φ )with C , C arbitrary constants. Therefore, α ( φ, ψ ) = ( φ, ψ ; (cid:112) J C , C (cid:112) f ( φ ) + J C r √ m sin φ, J C r √ m sin φ − C (cid:112) f ( φ ))is an 1-cocycle of the skew-symmetric algebroid ¯ D → T , for all ( C , C , C ) ∈ R and moreoverit satisfies Equations (5.25). Hence, we can use Corollary 5.6 to obtain solutions of the reducedsnakeboard problem. First, we calculate the integral curves of the vector field ξ (¯ L, ¯ D ) α :˙ φ ( t ) = C ˙ ψ ( t ) = 1 (cid:112) f ( φ ( t )) (cid:18) C (cid:112) f ( φ ( t )) + J C r √ m sin φ ( t ) (cid:19) = C + J C r (cid:112) mf ( φ ( t )) sin φ ( t )whose solutions are: φ ( t ) = C t + C ψ ( t ) = C t − C C log (cid:20) √ (cid:18)(cid:112) J cos( C t + C ) + (cid:113) mr − J sin ( C t + C ) (cid:19)(cid:21) + C (if C (cid:54) = 0) ψ ( t ) = C t + √ J C t sin( C ) (cid:112) mr − J sin ( C ) + C (if C = 0) AMILTON-JACOBI EQUATION AND NONHOLONOMIC MECHANICS 39 for all constants C i ∈ R , 1 ≤ i ≤ 5. Now, by a direct application of the nonholonomic equation weobtain that v ( t ) = (cid:112) J C v ( t ) = C (cid:112) f ( C t + C ) + J C r √ m sin( C t + C ) v ( t ) = J C r √ m sin( C t + C ) − C (cid:112) f ( C t + C )are solutions of the reduced nonholonomic problem.6. Conclusions and Future Work In this paper we have elucidated the geometrical framework for the Hamilton-Jacobi equation.Our formalism is valid for nonholonomic mechanical systems. The basic geometric ingredients are avector bundle, a linear almost Poisson bracket and a Hamiltonian function both on the dual bundle.We also have discussed the behavior of the theory under Hamiltonian morphisms and its applicabilityto reduction theory. Some examples are studied in detail and, as a consequence, it is shown the utilityof our framework to integrate the dynamical equations. However, in this direction more work mustbe done.In particular, as a future research, we will study new particular examples, testing candidates forsolutions of the nonholonomic Hamilton-Jacobi equation of the form α = d D f , for some f ∈ C ∞ ( Q )(if there exists) and moreover we will study the complete solutions for the Hamilton-Jacobi equationusing the groupoid theory. In this line, we will study the construction of numerical integrators viaHamilton-Jacobi theory [18]. We will also discuss the extension of our formalism to time-dependentLagrangian systems subjected to affine constraints in the velocities. It would be interesting to describethe Hamilton-Jacobi theory for variational constrained problems, giving a geometric interpretationof the Hamilton-Jacobi-Bellman equation for optimal control systems. Finally, extensions to classicalfield theories in the present context could be developed. Appendix Let {· , ·} D ∗ be a linear almost Poisson structure on a vector bundle τ D : D → Q , ([[ · , · ]] D , ρ D ) bethe corresponding skew-symmetric Lie algebroid structure on D and α : Q → D ∗ be a section of τ D ∗ : D ∗ → Q . If q ∈ Q then we may choose local coordinates ( q U ) = ( q i , q a ) on an open subset U of Q , q ∈ U , and a basis of sections { X A } = { X i , X γ } of the vector bundle τ − D ( U ) → U such that ρ D ( X i )( q ) = ∂∂q i | q , ρ D ( X γ )( q ) = 0 . (A.1)Suppose that ρ D ( X A ) = ρ UA ∂∂q U , [[ X A , X B ]] D = C CAB X C (A.2)and that the local expression of α in U is α ( q U ) = ( q U , α A ( q U )) . (A.3) Denote by Λ D ∗ the linear almost Poisson 2-vector on D ∗ and by ( q U , p A ) = ( q i , q a , p i , p γ ) the corre-sponding local coordinates on D ∗ . Then, from (2.1), it follows thatΛ D ∗ ( α ( q )) = ∂∂q i | α ( q ) ∧ ∂∂p i | α ( q ) − C CAB ( q ) α C ( q ) ∂∂p A | α ( q ) ∧ ∂∂p B | α ( q ) . (A.4)Moreover, using (A.1), (A.2) and (A.3), we obtain that( d D α )( q )( X i ( q ) , X j ( q )) = ∂α j ∂q i | q − ∂α i ∂q j | q − C Aij ( q ) α A ( q ) , ( d D α )( q )( X i ( q ) , X γ ( q )) = ∂α γ ∂q i | q − C Aiγ ( q ) α A ( q ) , ( d D α )( q )( X γ ( q ) , X ν ( q )) = − C Aγν ( q ) α A ( q ) . (A.5)On the other hand, let L α,D ( q ) be the subspace of T α ( q ) D ∗ defined by (4.3). Then, from (A.1) and(A.3), we deduce that L α,D ( q ) = (cid:104){ ∂∂q i | α ( q ) + ∂α A ∂q i | q ∂∂p A | α ( q ) }(cid:105) (A.6)which implies that( L α,D ( q )) = (cid:104){ dq a ( α ( q )) , dp j ( α ( q )) − ∂α j ∂q i | q dq i ( α ( q )) , dp γ ( α ( q )) − ∂α γ ∂q i | q dq i ( α ( q )) }(cid:105) . (A.7)In addition, using (A.4), one may prove that Λ D ∗ ( dq a ( α ( q ))) = 0 , Λ D ∗ ( dp j ( α ( q )) − ∂α j ∂q i | q dq i ( α ( q ))) = − ∂∂q j | α ( q ) − ( ∂α j ∂q i | q − C Cij ( q ) α C ( q )) ∂∂p i | α ( q ) − C Cjγ ( q ) α C ( q ) ∂∂p γ | α ( q ) , Λ D ∗ ( dp γ ( α ( q )) − ∂α γ ∂q i | q dq i ( α ( q ))) = − ( ∂α γ ∂q i | q − C Ciγ ( q ) α C ( q )) ∂∂p i | α ( q ) − C Cγν ( q ) α C ( q ) ∂∂p ν | α ( q ) . (A.8) Proof of Proposition 4.3. From (A.5), (A.6), (A.7) and (A.8), we deduce the result. (cid:3) Proof of Proposition 4.5. Suppose that β α ( q ) = λ U dq U ( α ( q )) + µ A dp A ( α ( q )) ∈ T ∗ α ( q ) D ∗ . Then, using (A.4) and (A.5), it follows that β α ( q ) ∈ Ker Λ D ∗ ( α ( q )) ⇐⇒ µ i = 0 , λ i = − ∂α γ ∂q i | q µ γ , for all i. Thus, from (A.7), we conclude that β α ( q ) ∈ ( L α,D ( q )) . 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Math. Phys. (1999), 545–560. M. de Le´on: Instituto de Ciencias Matem´aticas (CSIC-UAM-UC3M-UCM), Serrano 123, 28006 Madrid, Spain E-mail address : [email protected] Juan C. Marrero: ULL-CSIC Geometr´ıa Diferencial y Mec´anica Geom´etrica, Departamento de Matem´atica Fundamental,Facultad de Matem´aticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands, Spain E-mail address : [email protected] D. Mart´ın de Diego: Instituto de Ciencias Matem´aticas (CSIC-UAM-UC3M-UCM), Serrano 123, 28006 Madrid, Spain E-mail address ::